The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ of Proposition 7.5.9.1.1 is uniquely determined by the following requirements:
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1.
Existence of an Internal Hom. The tensor product
\[ \otimes _{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*} \]of $\mathsf{Sets}_{*}$ admits an internal Hom $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$.
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The Unit Object Is $S^{0}$. We have $\mathbb {1}_{\mathsf{Sets}_{*}}\cong S^{0}$.
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Let $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.
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2.
Since $\otimes _{\mathsf{Sets}_{*}}$ is part of a closed monoidal structure, it preserves colimits in each variable by
.
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3.
Since $X\cong \bigvee _{x\in X^{-}}S^{0}$ and $\otimes _{\mathsf{Sets}_{*}}$ preserves colimits in each variable, we have
\begin{align*} X\otimes _{\mathsf{Sets}_{*}}Y & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\otimes _{\mathsf{Sets}_{*}}Y\\ & \cong \bigvee _{x\in X^{-}}\webleft (S^{0}\otimes _{\mathsf{Sets}_{*}}Y\webright )\\ & \cong \bigvee _{x\in X^{-}}Y\\ & \cong \bigvee _{x\in X^{-}}S^{0}\wedge Y\\ & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\wedge Y\\ & \cong X\wedge Y, \end{align*}naturally in $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, where we have used that $S^{0}$ is the monoidal unit for $\otimes _{\mathsf{Sets}_{*}}$. Thus $X\otimes _{\mathsf{Sets}_{*}}-\cong X\wedge -$ for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.
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4.
Similarly, $-\otimes _{\mathsf{Sets}_{*}}Y\cong -\wedge Y$ for each $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.
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5.
By
, we then have $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$.
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Subdiagrams $\webleft (1\webright )$, $\webleft (2\webright )$, and $\webleft (3\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (4\webright )$ commutes by
.
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*},-1}$.
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Subdiagram $\webleft (6\webright )$ commutes trivially.
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Subdiagram $\webleft (7\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*}}$, where the equality $\rho ^{\mathsf{Sets}_{*}}_{S^{0}}=\lambda ^{\mathsf{Sets}_{*}}_{S^{0}}$ comes from
.
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\lambda ^{\prime ,-1}$.
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Subdiagram $\webleft (\ddagger \webright )$ commutes, as proved above.
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Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\lambda ^{\mathsf{Sets}_{*},-1}$.
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;
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Subdiagrams $\webleft (2\webright )$ and $\webleft (3\webright )$ commute by the functoriality of $\otimes $;
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Subdiagram $\webleft (4\webright )$ commutes by the left monoidal unity of $\webleft (\operatorname {\mathrm{id}}^{\otimes },\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\webright )$, which we proved above;
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\lambda '$;
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Subdiagram $\webleft (6\webright )$ commutes by the naturality of $\rho '$, where the equality $\rho '_{\mathbb {1}_{\mathsf{Sets}_{*}}}=\lambda '_{\mathbb {1}_{\mathsf{Sets}_{*}}}$ comes from
;
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Subdiagrams $\webleft (1\webright )$ and $\webleft (6\webright )$ commute by
;
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Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (\S \webright )$ commutes, as was shown above;
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Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\lambda ^{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (4\webright )$ commutes trivially;
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Subdiagram $\webleft (5\webright )$ commutes by Chapter 12: Constructions With Monoidal Categories, Item 2c of Item 2 of Proposition 12.1.1.1.4, whose proof uses only the left monoidal unity of $\webleft (\operatorname {\mathrm{id}}^{\otimes },\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\webright )$, which has been proven above;
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\sigma '$ and the fact that $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}$ is invertible;
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Subdiagram $\webleft (\ddagger \webright )$ commutes as proved above;
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Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\sigma ^{\mathsf{Sets}_{*}}$ and the fact that $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}$ is invertible;
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Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\sigma ^{\mathsf{Sets}_{*},-1}$.
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}$.
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Subdiagram $\webleft (¶\webright )$ commutes, as proved above.
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Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\sigma ^{\prime ,-1}$.
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}$.
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Subdiagram $\webleft (1\webright )$ commutes by the braidedness of $\operatorname {\mathrm{id}}^{\otimes }$, as proved above.
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Subdiagram $\webleft (2\webright )$ commutes by
.
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Subdiagrams $\webleft (1\webright )$, $\webleft (2\webright )$, and $\webleft (3\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$.
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Subdiagrams $\webleft (4\webright )$, $\webleft (5\webright )$, and $\webleft (6\webright )$ commute by the naturality of $\lambda ^{\mathsf{Sets}_{*}}$, where the equality $\lambda ^{\mathsf{Sets}_{*}}_{S^{0}}=\rho ^{\mathsf{Sets}_{*}}_{S^{0}}$ comes from
.
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\rho ^{\prime ,-1}$.
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Subdiagram $\webleft (\dagger \webright )$ commutes, as proved above.
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Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*},-1}$.
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Subdiagrams $\webleft (1\webright )$, $\webleft (4\webright )$, $\webleft (5\webright )$, $\webleft (8\webright )$, and $\webleft (11\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (2\webright )$ commutes by the right monoidal unity of $\webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )$;
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Subdiagram $\webleft (3\webright )$ commutes by the triangle identity for $\webleft (\alpha ',\lambda ',\rho '\webright )$;
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Subdiagram $\webleft (6\webright )$ commutes by
;
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Subdiagram $\webleft (7\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (9\webright )$ commutes by
;
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Subdiagram $\webleft (10\webright )$ commutes by
;
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Subdiagrams $\webleft (1\webright )$, $\webleft (3\webright )$, $\webleft (4\webright )$, and $\webleft (6\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{Sets_{*}}$;
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Subdiagram $\webleft (\ddagger \webright )$ commutes, as proved above;
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Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\alpha '$;
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\alpha ^{\mathsf{Sets}_{*}}$;
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\alpha ^{\mathsf{Sets}_{*},-1}$.
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Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (\dagger \webright )$ commutes, as proved above.
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Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.
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Subdiagram $\webleft (6\webright )$ commutes by the naturality of $\alpha ^{\prime ,-1}$.
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1.
Two-Sided Preservation of Colimits. The tensor product
\[ \otimes _{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*} \]of $\mathsf{Sets}_{*}$ preserves colimits separately in each variable.
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2.
The Unit Object Is $S^{0}$. We have $\mathbb {1}_{\mathsf{Sets}_{*}}\cong S^{0}$.
7.5.10 The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
More precisely, the full subcategory of the category $\mathcal{M}^{\mathrm{cld}}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of spanned by the closed symmetric monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ satisfying Item 1 and Item 2 is contractible (i.e. equivalent to the punctual category).
Proof of Theorem 7.5.10.1.1.
Unwinding the Statement
Let $\webleft (\mathsf{Sets}_{*},\otimes _{\mathsf{Sets}_{*}},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}},\mathbb {1}_{\mathsf{Sets}_{*}},\lambda ',\rho ',\sigma '\webright )$ be a closed symmetric monoidal category satisfying Item 1 and Item 2. We need to show that the identity functor
\[ \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]
admits a unique closed symmetric monoidal functor structure
\[ \begin{array}{cccc} \phantom{\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}}\mathllap {\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}} \colon \mkern -10mu & X\otimes _{\mathsf{Sets}_{*}}Y \mkern -10mu& {}\mathbin {\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}& \mkern -10mu{}X\wedge Y,\\ \phantom{\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}}\mathllap {\operatorname {\mathrm{id}}^{\operatorname {\mathrm{Hom}}}_{\mathsf{Sets}_{*}}} \colon \mkern -10mu & \webleft [X,Y\webright ]_{\mathsf{Sets}_{*}} \mkern -10mu& {}\mathbin {\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}& \mkern -10mu{}\mathsf{Sets}_{*}\webleft (X,Y\webright ),\\ \phantom{\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}}\mathllap {\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}} \colon \mkern -10mu & \mathbb {1}_{\mathsf{Sets}_{*}} \mkern -10mu& {}\mathbin {\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}& \mkern -10mu{}S^{0}, \end{array} \]
making it into a symmetric monoidal strongly closed isomorphism of categories from $\left(\phantom{\mathrlap {\lambda '}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda '$, $\rho '$, $\left.\sigma '\right)$ to the closed symmetric monoidal category $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\times $, $\mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ of Proposition 7.5.9.1.1.
Constructing an Isomorphism $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}\cong \mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )$
By , we have a natural isomorphism
\[ \mathsf{Sets}_{*}\webleft (S^{0},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}\webright )\cong \mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright ). \]
By Chapter 6: Pointed Sets, Item 4 of Proposition 6.1.4.1.1, we also have a natural isomorphism
\[ \mathsf{Sets}_{*}\webleft (S^{0},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}\webright )\cong \webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}. \]
Composing both natural isomorphisms, we obtain a natural isomorphism
\[ \mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )\cong \webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}. \]
Given $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, we will write
\[ \operatorname {\mathrm{id}}^{\operatorname {\mathrm{Hom}}}_{X,Y}\colon \mathsf{Sets}_{*}\webleft (X,Y\webright )\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft [X,Y\webright ]_{\mathsf{Sets}_{*}} \]
for the component of this isomorphism at $\webleft (X,Y\webright )$.
Constructing an Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
Since $\otimes _{\mathsf{Sets}_{*}}$ is adjoint in each variable to $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$ by assumption and $\wedge $ is adjoint in each variable to $\mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )$ by Chapter 4: Constructions With Sets, Item 2 of Proposition 4.3.5.1.2, uniqueness of adjoints () gives us natural isomorphisms
\begin{align*} X\otimes _{\mathsf{Sets}_{*}}- & \cong X\wedge -,\\ -\otimes _{\mathsf{Sets}_{*}}Y & \cong Y\wedge -. \end{align*}
By , we then have $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$. We will write
\[ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}\colon X\otimes _{\mathsf{Sets}_{*}}Y\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X\wedge Y \]
for the component of this isomorphism at $\webleft (X,Y\webright )$.
Alternative Construction of an Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
Alternatively, we may construct a natural isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$ as follows:
Below, we’ll show that if a natural isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$ exists, then it must be unique. This will show that the isomorphism constructed above is equal to the isomorphism $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}\colon X\otimes _{\mathsf{Sets}_{*}}Y\to X\wedge Y$ from before.
Constructing an Isomorphism $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\colon \mathbb {1}_{\mathsf{Sets}_{*}}\to S^{0}$
We define an isomorphism $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\colon \mathbb {1}_{\mathsf{Sets}_{*}}\to S^{0}$ as the composition
\[ \mathbb {1}_{\mathsf{Sets}_{*}}\overset {\rho ^{\mathsf{Sets}_{*},-1}_{\mathbb {1}_{\mathsf{Sets}_{*}}}}{\underset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}\mathbb {1}_{\mathsf{Sets}_{*}}\wedge S^{0}\overset {\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|\mathbb {1}_{\mathsf{Sets}_{*}}}}{\underset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}\mathbb {1}_{\mathsf{Sets}_{*}}\otimes _{\mathsf{Sets}_{*}} S^{0}\overset {\lambda '_{ S^{0}}}{\underset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}S^{0} \]
in $\mathsf{Sets}_{*}$.
Monoidal Left Unity of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
We have to show that the diagram 
Since all subdiagrams commute, so does the boundary diagram, i.e. the diagram $\webleft (\dagger \webright )$ above. As a result, the diagram
it follows that the diagram
Here’s a step-by-step showcase of this argument: 
We then have
\begin{align*} \lambda ^{\prime ,-1}_{X}\webleft (x\webright ) & = \webleft [\lambda ^{\prime ,-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webright ]\webleft (x\webright ) \end{align*}
for each $x\in X$, and thus we have
\[ \lambda ^{\prime ,-1}_{X}=\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}. \]
Taking inverses then gives
\[ \lambda ^{\prime }_{X}=\lambda ^{\mathsf{Sets}_{*}}_{X}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|S^{0},X}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright ), \]
showing that the diagram
Braidedness of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
We have to show that the diagram 
it follows that the boundary diagram, i.e. diagram $\webleft (\S \webright )$, also commutes. Next, consider the diagram
it follows that the boundary diagram, i.e. diagram $\webleft (\ddagger \webright )$, also commutes. Next, consider the diagram
it follows that the boundary diagram, i.e. diagram $\webleft (\dagger \webright )$ also commutes. Taking inverses for the diagram $\webleft (\dagger \webright )$, we see that the diagram
it follows that the diagram
\begin{align*} \webleft [\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\circ \sigma ^{\mathsf{Sets}_{*},-1}_{X,Y}\webright ]\webleft (y,x\webright ) & = \webleft [\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\circ \sigma ^{\mathsf{Sets}_{*},-1}_{X,Y}\circ \webleft (\webleft [y\webright ]\wedge \webleft [x\webright ]\webright )\webright ]\webleft (1,1\webright )\\ & = \webleft [\sigma ^{\prime ,-1}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,X}\circ \webleft (\webleft [y\webright ]\wedge \webleft [x\webright ]\webright )\webright ]\webleft (1,1\webright )\\ & = \webleft [\sigma ^{\prime ,-1}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,X}\webright ]\webleft (y,x\webright ) \end{align*}
for each $\webleft (y,x\webright )\in Y\wedge X$, and thus we have
\[ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\circ \sigma ^{\mathsf{Sets}_{*},-1}_{X,Y}=\sigma ^{\prime ,-1}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,X}. \]
Taking inverses then gives
\[ \sigma ^{\mathsf{Sets}_{*}}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}=\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|Y,X}\circ \sigma ^{\prime }_{X,Y}, \]
showing that the diagram
Monoidal Right Unity of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
We have to show that the diagram 
\[ \sigma '_{S^{0},\mathbb {1}_{\mathsf{Sets}_{*}}}\colon S^{0}\otimes _{\mathsf{Sets}_{*}}\mathbb {1}_{\mathsf{Sets}_{*}}\to \mathbb {1}_{\mathsf{Sets}_{*}}\otimes _{\mathsf{Sets}_{*}}S^{0} \]
as the composition
Next, consider the diagram
. In this diagram:
Since all subdiagrams commute, so does the boundary diagram, i.e. the diagram $\webleft (\dagger \webright )$ above. As a result, the diagram
it follows that the diagram
Here’s a step-by-step showcase of this argument: 
We then have
\begin{align*} \rho ^{\prime ,-1}_{X}\webleft (a\webright ) & = \webleft [\rho ^{\prime ,-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \rho ^{\mathsf{Sets}_{*},-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \rho ^{\mathsf{Sets}_{*},-1}_{X}\webright ]\webleft (a\webright ) \end{align*}
for each $a\in X$, and thus we have
\[ \rho ^{\prime ,-1}_{X}=\webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \rho ^{\mathsf{Sets}_{*},-1}_{X}. \]
Taking inverses then gives
\[ \rho ^{\prime }_{X}=\rho ^{\mathsf{Sets}_{*}}_{X}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|S^{0},X}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}\webright ), \]
showing that the diagram
Monoidality of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
We have to show that the diagram 
it follows that the boundary diagram, i.e. diagram $\webleft (\ddagger \webright )$, also commutes. Consider now the diagram
it follows that the boundary diagram, i.e. diagram $\webleft (\dagger \webright )$, also commutes. Taking inverses on the diagram $\webleft (\dagger \webright )$, we see that the diagram
it follows that the diagram
\begin{align*} \left[\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X\wedge Y,Z}\right.\\ \left.{}\circ \alpha ^{\mathsf{Sets}_{*},-1}_{X,Y,Z}\right]\webleft (x,\webleft (y,z\webright )\webright ) & = \left[\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X\wedge Y,Z}\right.\\ & \phantom{={}} \mkern 4mu\left.{}{}\circ \alpha ^{\mathsf{Sets}_{*},-1}_{X,Y,Z}\circ \webleft (\webleft [x\webright ]\wedge \webleft (\webleft [y\webright ]\wedge \webleft [z\webright ]\webright )\webright )\right]\webleft (1,\webleft (1,1\webright )\webright )\\ & = \left[\alpha ^{\prime ,-1}_{X,Y,Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,Z}\webright )\right.\\ & \phantom{={}} \mkern 4mu\left.{}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y\wedge Z}\circ \webleft (\webleft [x\webright ]\wedge \webleft (\webleft [y\webright ]\wedge \webleft [z\webright ]\webright )\webright )\right]\webleft (1,\webleft (1,1\webright )\webright )\\ & = \webleft [\alpha ^{\prime ,-1}_{X,Y,Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y\wedge Z}\webright ]\webleft (x,\webleft (y,z\webright )\webright ) \end{align*}
\[ \webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X\wedge Y,Z}\circ \alpha ^{\mathsf{Sets}_{*},-1}_{X,Y,Z}=\alpha ^{\prime ,-1}_{X,Y,Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y\wedge Z}. \]
\[ \alpha ^{\mathsf{Sets}_{*}}_{X,Y,Z}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X\wedge Y,Z}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )=\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y\wedge Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|Y,Z}\webright )\circ \alpha ^{\prime }_{X,Y,Z}, \]
Uniqueness of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
Let $\phi ,\psi \colon -_{1}\otimes _{\mathsf{Sets}_{*}}-_{2}\Rightarrow -_{1}\wedge -_{2}$ be natural isomorphisms. Since these isomorphisms are compatible with the unitors of $\mathsf{Sets}_{*}$ with respect to $\wedge $ and $\otimes $ (as shown above), we have
\begin{align*} \lambda '_{Y} & = \lambda ^{\mathsf{Sets}_{*}}_{Y}\circ \phi _{S^{0},Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ),\\ \lambda '_{Y} & = \lambda ^{\mathsf{Sets}_{*}}_{Y}\circ \psi _{S^{0},Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ). \end{align*}
Postcomposing both sides with $\lambda ^{\mathsf{Sets}_{*},-1}_{Y}$ and then precomposing both sides with $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}$ gives
\begin{align*} \lambda ^{\mathsf{Sets}_{*},-1}_{Y}\circ \lambda '_{Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ) & = \phi _{S^{0},Y},\\ \lambda ^{\mathsf{Sets}_{*},-1}_{Y}\circ \lambda '_{Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ) & = \psi _{S^{0},Y}, \end{align*}
and thus we have
\[ \phi _{S^{0},Y}=\psi _{S^{0},Y} \]
for each $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$. Now, let $x\in X$ and consider the naturality diagrams
\begin{align*} \phi _{X,Y}\webleft (x,y\webright ) & = \webleft [\phi _{X,Y}\circ \webleft (\webleft [x\webright ]\wedge \operatorname {\mathrm{id}}_{Y}\webright )\webright ]\webleft (1,y\webright )\\ & = \webleft [\webleft (\webleft [x\webright ]\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Y}\webright )\circ \phi _{S^{0},Y}\webright ]\webleft (1,y\webright )\\ & = \webleft [\webleft (\webleft [x\webright ]\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Y}\webright )\circ \psi _{S^{0},Y}\webright ]\webleft (1,y\webright )\\ & = \webleft [\psi _{X,Y}\circ \webleft (\webleft [x\webright ]\wedge \operatorname {\mathrm{id}}_{Y}\webright )\webright ]\webleft (1,y\webright )\\ & = \psi _{X,Y}\webleft (x,y\webright ) \end{align*}
for each $\webleft (x,y\webright )\in X\wedge Y$. Therefore we have
\[ \phi _{X,Y}=\psi _{X,Y} \]
for each $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$ and thus $\phi =\psi $, showing the isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\times }$ to be unique.
The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ of Proposition 7.5.9.1.1 is uniquely determined by the following requirements:
More precisely, the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of spanned by the symmetric monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ satisfying Item 1 and Item 2 is contractible.
Proof of Corollary 7.5.10.1.2.
Since $\mathsf{Sets}_{*}$ is locally presentable (), it follows from that Corollary 7.5.10.1.2 is equivalent to the existence of an internal Hom as in Item 1 of Theorem 7.5.10.1.1. The result then follows from Theorem 7.5.10.1.1.
The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ is the unique symmetric monoidal structure on $\mathsf{Sets}_{*}$ such that the free pointed set functor
\[ \webleft (-\webright )^{+} \colon \mathsf{Sets}\to \mathsf{Sets}_{*} \]
admits a symmetric monoidal structure, i.e. the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of spanned by the symmetric monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ with respect to which $\webleft (-\webright )^{+}$ admits a symmetric monoidal structure is contractible.
Proof of Corollary 7.5.10.1.3.
Let $\webleft (\otimes _{\mathsf{Sets}_{*}},\mathbb {1}_{\mathsf{Sets}_{*}},\lambda ^{\mathsf{Sets}_{*}},\rho ^{\mathsf{Sets}_{*}},\sigma ^{\mathsf{Sets}_{*}}\webright )$ be a symmetric monoidal structure on $\mathsf{Sets}_{*}$ such that $\webleft (-\webright )^{+}$ admits a symmetric monoidal structure with respect to $\otimes _{\mathsf{Sets}_{*}}$ and $\wedge $. We have isomorphisms
\begin{align*} X\otimes _{\mathsf{Sets}_{*}}Y & \cong \webleft (X^{-}\webright )^{+}\otimes _{\mathsf{Sets}_{*}}\webleft (Y^{-}\webright )^{+}\\ & \cong \webleft (X^{-}\times Y^{-}\webright )^{+}\\ & \cong \webleft (X^{-}\webright )^{+}\wedge \webleft (Y^{-}\webright )^{+}\\ & \cong X\wedge Y, \end{align*}
all natural in $X$ and $Y$. Now, since $\wedge $ preserves colimits in both variables and $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$, it follows that $\otimes _{\mathsf{Sets}_{*}}$ also preserves colimits in both variables, so the result then follows from Corollary 7.5.10.1.2.
