Let $X$ be a set.
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1.
The discrete category on $X$ is the category $X_{\mathsf{disc}}$ where
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Objects. We have
\[ \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{disc}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X. \] -
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Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{disc}}\webright )$, we have
\[ \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}\webleft (A,B\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \operatorname {\mathrm{id}}_{A} & \text{if $A=B$,}\\ \text{Ø}& \text{if $A\neq B$.} \end{cases} \] -
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Identities. For each $A\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{disc}}\webright )$, the unit map
\[ \mathbb {1}^{X_{\mathsf{disc}}}_{A} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}\webleft (A,A\webright ) \]of $X_{\mathsf{disc}}$ at $A$ is defined by
\[ \operatorname {\mathrm{id}}^{X_{\mathsf{disc}}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{A}. \] -
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Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (X_{\mathsf{disc}}\webright )$, the composition map
\[ \circ ^{X_{\mathsf{disc}}}_{A,B,C} \colon \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}\webleft (B,C\webright ) \times \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{X_{\mathsf{disc}}}\webleft (A,C\webright ) \]of $X_{\mathsf{disc}}$ at $\webleft (A,B,C\webright )$ is defined by
\[ \operatorname {\mathrm{id}}_{A}\circ \operatorname {\mathrm{id}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{A}. \]
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2.
A category $\mathcal{C}$ is discrete if it is equivalent to $X_{\mathsf{disc}}$ for some set $X$.
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Functoriality. The assignment $X\mapsto X_{\mathsf{disc}}$ defines a functor
\[ \webleft (-\webright )_{\mathsf{disc}} \colon \mathsf{Sets}\to \mathsf{Cats}. \] -
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Adjointness. We have a quadruple adjunction
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3.
Symmetric Strong Monoidality With Respect to Coproducts. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )_{\mathsf{disc}},\webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}},\webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}\webright ) \to \webleft (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}\webright ), \]being equipped with isomorphisms
\[ \begin{gathered} \webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y_{\mathsf{disc}} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )_{\mathsf{disc}},\\ \webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}} \colon \text{Ø}_{\mathsf{cat}}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø}_{\mathsf{disc}}, \end{gathered} \]natural in $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
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4.
Symmetric Strong Monoidality With Respect to Products. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )_{\mathsf{disc}},\webleft (-\webright )^{\times }_{\mathsf{disc}},\webleft (-\webright )^{\times }_{\mathsf{disc}|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright ) \to \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ), \]being equipped with isomorphisms
\[ \begin{gathered} \webleft (-\webright )^{\times }_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\times Y_{\mathsf{disc}} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\times Y\webright )_{\mathsf{disc}},\\ \webleft (-\webright )^{\times }_{\mathsf{disc}|\mathbb {1}} \colon \mathsf{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{pt}_{\mathsf{disc}}, \end{gathered} \]natural in $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
Let $X$ be a set.
Proof of Proposition 11.3.3.1.2.
Item 1: Functoriality
Clear.
Item 2: Adjointness
This is proved in Proposition 11.3.1.1.1.
Item 3: Symmetric Strong Monoidality With Respect to Coproducts
Clear.
Item 4: Symmetric Strong Monoidality With Respect to Products
Clear.
