Let $A$, $B$, $C$, and $X$ be sets.
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Functoriality. The assignments $A,B,\webleft (A,B\webright )\mapsto A\times B$ define functors
\begin{gather*} \begin{aligned} A\times - & \colon \mathsf{Sets}\to \mathsf{Sets},\\ -\times B & \colon \mathsf{Sets}\to \mathsf{Sets}, \end{aligned}\\ -_{1}\times -_{2} \colon \mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets}, \end{gather*}
where $-_{1}\times -_{2}$ is the functor where
- Action on Objects. For each $\webleft (A,B\webright )\in \text{Obj}\webleft (\mathsf{Sets}\times \mathsf{Sets}\webright )$, we have
\[ \webleft [-_{1}\times -_{2}\webright ]\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A\times B. \]
- Action on Morphisms. For each $\webleft (A,B\webright ),\webleft (X,Y\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on $\textup{Hom}$-sets
\[ \times _{\webleft (A,B\webright ),\webleft (X,Y\webright )} \colon \mathsf{Sets}\webleft (A,X\webright )\times \mathsf{Sets}\webleft (B,Y\webright )\to \mathsf{Sets}\webleft (A\times B,X\times Y\webright ) \]
of $\times $ at $\webleft (\webleft (A,B\webright ),\webleft (X,Y\webright )\webright )$ is defined by sending $\webleft (f,g\webright )$ to the function
\[ f\times g\colon A\times B\to X\times Y \]defined by
\[ \webleft [f\times g\webright ]\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f\webleft (a\webright ),g\webleft (b\webright )\webright ) \]for each $\webleft (a,b\webright )\in A\times B$.
- Action on Objects. For each $\webleft (A,B\webright )\in \text{Obj}\webleft (\mathsf{Sets}\times \mathsf{Sets}\webright )$, we have
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Adjointness. We have adjunctions witnessed by bijections
\begin{align*} \textup{Hom}_{\mathsf{Sets}}\webleft (A\times B,C\webright ) & \cong \textup{Hom}_{\mathsf{Sets}}\webleft (A,\textup{Hom}_{\mathsf{Sets}}\webleft (B,C\webright )\webright ),\\ \textup{Hom}_{\mathsf{Sets}}\webleft (A\times B,C\webright ) & \cong \textup{Hom}_{\mathsf{Sets}}\webleft (B,\textup{Hom}_{\mathsf{Sets}}\webleft (A,C\webright )\webright ), \end{align*}
natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Associativity. We have an isomorphism of sets
\[ \webleft (A\times B\webright )\times C \cong A\times \webleft (B\times C\webright ), \]
natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Unitality. We have isomorphisms of sets
\begin{align*} \text{pt}\times A & \cong A,\\ A\times \text{pt}& \cong A, \end{align*}
natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Commutativity. We have an isomorphism of sets
\[ A\times B \cong B\times A, \]
natural in $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Annihilation With the Empty Set. We have isomorphisms of sets
\begin{align*} A\times \emptyset & \cong \emptyset ,\\ \emptyset \times A & \cong \emptyset , \end{align*}
natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Distributivity Over Unions. We have isomorphisms of sets
\begin{align*} A\times \webleft (B\cup C\webright ) & = \webleft (A\times B\webright )\cup \webleft (A\times C\webright ),\\ \webleft (A\cup B\webright )\times C & = \webleft (A\times C\webright )\cup \webleft (B\times C\webright ). \end{align*}
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Distributivity Over Intersections. We have isomorphisms of sets
\begin{align*} A\times \webleft (B\cap C\webright ) & = \webleft (A\times B\webright )\cap \webleft (A\times C\webright ),\\ \webleft (A\cap B\webright )\times C & = \webleft (A\times C\webright )\cap \webleft (B\times C\webright ). \end{align*}
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Middle-Four Exchange with Respect to Intersections. We have an isomorphism of sets
\[ \webleft (A\times B\webright )\cap \webleft (C\times D\webright )\cong \webleft (A\cap B\webright )\times \webleft (C\cap D\webright ). \]
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Distributivity Over Differences. We have isomorphisms of sets
\begin{align*} A\times \webleft (B\setminus C\webright ) & = \webleft (A\times B\webright )\setminus \webleft (A\times C\webright ),\\ \webleft (A\setminus B\webright )\times C & = \webleft (A\times C\webright )\setminus \webleft (B\times C\webright ), \end{align*}
natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Distributivity Over Symmetric Differences. We have isomorphisms of sets
\begin{align*} A\times \webleft (B\mathbin {\triangle }C\webright ) & = \webleft (A\times B\webright )\mathbin {\triangle }\webleft (A\times C\webright ),\\ \webleft (A\mathbin {\triangle }B\webright )\times C & = \webleft (A\times C\webright )\mathbin {\triangle }\webleft (B\times C\webright ), \end{align*}
natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
- Symmetric Monoidality. The triple $\webleft (\mathsf{Sets},\times ,\text{pt}\webright )$ is a symmetric monoidal category.
- Symmetric Bimonoidality. The quintuple $\webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset ,\times ,\text{pt}\webright )$ is a symmetric bimonoidal category.