Let $X$ be a set.
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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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Symmetric Strong Monoidality With Respect to Coproducts. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )_{\mathsf{disc}},\webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}},\webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset \webright ) \to \webleft (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset _{\mathsf{cat}}\webright ), \]
being equipped with isomorphisms
\[ \begin{gathered} \webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y_{\mathsf{disc}} \xrightarrow {\cong }\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )_{\mathsf{disc}},\\ \webleft (-\webright )^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathsf{disc}|\mathbb {1}} \colon \emptyset _{\mathsf{cat}}\xrightarrow {\cong }\emptyset _{\mathsf{disc}}, \end{gathered} \]natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Symmetric Strong Monoidality With Respect to Products. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )_{\mathsf{disc}},\webleft (-\webright )^{\times }_{\mathsf{disc}},\webleft (-\webright )^{\times }_{\mathsf{disc}|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \to \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ), \]
being equipped with isomorphisms
\[ \begin{gathered} \webleft (-\webright )^{\times }_{\mathsf{disc}|X,Y} \colon X_{\mathsf{disc}}\times Y_{\mathsf{disc}} \xrightarrow {\cong }\webleft (X\times Y\webright )_{\mathsf{disc}},\\ \webleft (-\webright )^{\times }_{\mathsf{disc}|\mathbb {1}} \colon \mathsf{pt}\xrightarrow {\cong }\text{pt}_{\mathsf{disc}}, \end{gathered} \]natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
