Let $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, $\webleft (Z,z_{0}\webright )$, and $\webleft (A,a_{0}\webright )$ be pointed sets.
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Functoriality. The assignment $\webleft (X,Y,Z,f,g\webright )\mapsto X\mathbin {\textstyle \coprod _{f,Z,g}}Y$ defines a functor
\[ -_{1}\mathbin {\textstyle \coprod _{-_{3}}}-_{1}\colon \mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}\webright )\to \mathsf{Sets}_{*}, \]
where $\mathcal{P}$ is the category that looks like this:
In particular, the action on morphisms of $-_{1}\mathbin {\textstyle \coprod _{-_{3}}}-_{1}$ is given by sending a morphism
in $\mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}_{*}\webright )$ to the morphism of pointed sets
\[ \xi \colon \webleft (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0}\webright )\overset {\exists !}{\to }\webleft (X'\mathbin {\textstyle \coprod _{Z'}}Y',p'_{0}\webright ) \]given by
\[ \xi \webleft (p\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \phi \webleft (x\webright ) & \text{if $p=\webleft [\webleft (0,x\webright )\webright ]$},\\ \psi \webleft (y\webright ) & \text{if $p=\webleft [\webleft (1,y\webright )\webright ]$} \end{cases} \]for each $p\in X\mathbin {\textstyle \coprod _{Z}}Y$, which is the unique morphism of pointed sets making the diagram
commute.
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Associativity. Given a diagram
in $\mathsf{Sets}$, we have isomorphisms of pointed sets
\[ \webleft (X\mathbin {\textstyle \coprod _{W}}Y\webright )\mathbin {\textstyle \coprod _{V}}Z\cong \webleft (X\mathbin {\textstyle \coprod _{W}}Y\webright )\mathbin {\textstyle \coprod _{Y}}\webleft (Y\mathbin {\textstyle \coprod _{V}}Z\webright ) \cong X\mathbin {\textstyle \coprod _{W}}\webleft (Y\mathbin {\textstyle \coprod _{V}}Z\webright ), \]where these pullbacks are built as in the diagrams
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Unitality. We have isomorphisms of sets
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Commutativity. We have an isomorphism of sets
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Interaction With Coproducts. We have
- Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{*},\mathbin {\textstyle \coprod _{X}},\webleft (X,x_{0}\webright )\webright )$ is a symmetric monoidal category.
