The product of $(X,x_{0})$ and $(Y,y_{0})$ is the product of $(X,x_{0})$ and $(Y,y_{0})$ in $\mathsf{Sets}_{*}$ as in 
, .
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The Limit. The pointed set $(X\times Y,(x_{0},y_{0}))$.
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The Cone. The morphisms of pointed sets
\begin{align*} \operatorname {\mathrm{\mathrm{pr}}}_{1} & \colon (X\times Y,(x_{0},y_{0}))\to (X,x_{0}),\\ \operatorname {\mathrm{\mathrm{pr}}}_{2} & \colon (X\times Y,(x_{0},y_{0}))\to (Y,y_{0}) \end{align*}defined by
\begin{align*} \operatorname {\mathrm{\mathrm{pr}}}_{1}(x,y) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x,\\ \operatorname {\mathrm{\mathrm{pr}}}_{2}(x,y) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}y \end{align*}for each $(x,y)\in X\times Y$.
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Functoriality. The assignments
\[ (X,x_{0}),(Y,y_{0}),((X,x_{0}),(Y,y_{0}))\mapsto (X\times Y,(x_{0},y_{0})) \]define functors
\[ \begin{array}{ccc} A\times -\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\times B\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\times -_{2}\colon \mkern -15mu & \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}, \end{array} \]defined in the same way as the functors of Chapter 4: Constructions With Sets, Item 1 of Proposition 4.1.3.1.3.
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Lack of Adjointness. The functors $X\times -$ and $-\times Y$ do not admit right adjoints.
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Associativity. We have an isomorphism of pointed sets
\[ ((X\times Y)\times Z,((x_{0},y_{0}),z_{0})) \cong (X\times (Y\times Z),(x_{0},(y_{0},z_{0}))) \]natural in $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$.
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Unitality. We have isomorphisms of pointed sets
\begin{align*} (\mathrm{pt},\star )\times (X,x_{0}) & \cong (X,x_{0}),\\ (X,x_{0})\times (\mathrm{pt},\star ) & \cong (X,x_{0}), \end{align*}natural in $(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$.
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Commutativity. We have an isomorphism of pointed sets
\[ (X\times Y,(x_{0},y_{0})) \cong (Y\times X,(y_{0},x_{0})), \]natural in $(X,x_{0}),(Y,y_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$.
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Symmetric Monoidality. The triple $(\mathsf{Sets}_{*},\times ,(\mathrm{pt},\star ))$ is a symmetric monoidal category.
6.2.3 Products
Let $(X,x_{0})$ and $(Y,y_{0})$ be pointed sets.
Concretely, the product of $(X,x_{0})$ and $(Y,y_{0})$ is the pair consisting of:
Proof of Construction 6.2.3.1.2.
We claim that $(X\times Y,(x_{0},y_{0}))$ is the categorical product of $(X,x_{0})$ and $(Y,y_{0})$ in $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form
\[ \phi \colon (P,*)\to (X\times Y,(x_{0},y_{0})) \]
making the diagram
\begin{align*} \operatorname {\mathrm{\mathrm{pr}}}_{1}\circ \phi & = p_{1},\\ \operatorname {\mathrm{\mathrm{pr}}}_{2}\circ \phi & = p_{2} \end{align*}
via
\[ \phi (x)=(p_{1}(x),p_{2}(x)) \]
for each $x\in P$. Note that this is indeed a morphism of pointed sets, as we have
\begin{align*} \phi (*) & = (p_{1}(*),p_{2}(*))\\ & = (x_{0},y_{0}),\end{align*}
where we have used that $p_{1}$ and $p_{2}$ are morphisms of pointed sets.
Let $(X,x_{0})$, $(Y,y_{0})$, and $(Z,z_{0})$ be pointed sets.
Proof of Proposition 6.2.3.1.3.
Item 1: Functoriality
This is a special case of functoriality of limits, , of .
Item 2: Lack of Adjointness
See [Yuan, Is the category of pointed sets Cartesian closed?].
Item 3: Associativity
This follows from Chapter 4: Constructions With Sets, Item 4 of Proposition 4.1.3.1.3.
Item 4: Unitality
This follows from Chapter 4: Constructions With Sets, Item 5 of Proposition 4.1.3.1.3.
Item 5: Commutativity
This follows from Chapter 4: Constructions With Sets, Item 6 of Proposition 4.1.3.1.3.
Item 6: Symmetric Monoidality
This follows from Chapter 4: Constructions With Sets, Item 14 of Proposition 4.1.3.1.3.
