Let $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, and $\webleft (Z,z_{0}\webright )$ be pointed sets.
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Functoriality. The assignments
\[ \webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )\mapsto \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \]
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 2: Constructions With Sets, Item 1 of Proposition 2.1.3.1.3.
- 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
\[ \webleft (\webleft (X\times Y\webright )\times Z,\webleft (\webleft (x_{0},y_{0}\webright ),z_{0}\webright )\webright ) \cong \webleft (X\times \webleft (Y\times Z\webright ),\webleft (x_{0},\webleft (y_{0},z_{0}\webright )\webright )\webright ) \]
natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
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Unitality. We have isomorphisms of pointed sets
\begin{align*} \webleft (\text{pt},\star \webright )\times \webleft (X,x_{0}\webright ) & \cong \webleft (X,x_{0}\webright ),\\ \webleft (X,x_{0}\webright )\times \webleft (\text{pt},\star \webright ) & \cong \webleft (X,x_{0}\webright ), \end{align*}
natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
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Commutativity. We have an isomorphism of pointed sets
\[ \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \cong \webleft (Y\times X,\webleft (y_{0},x_{0}\webright )\webright ), \]
natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
- Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{*},\times ,\webleft (\text{pt},\star \webright )\webright )$ is a symmetric monoidal category.
