Let $A$, $B$, $C$, and $X$ be sets.
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Functoriality. The assignment $\webleft (A,B,C,f,g\webright )\mapsto A\mathbin {\textstyle \coprod _{f,C,g}}B$ 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 map $\xi \colon A\mathbin {\textstyle \coprod _{C}}B\overset {\exists !}{\to }A'\mathbin {\textstyle \coprod _{C'}}B'$ given by
\[ \xi \webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \phi \webleft (a\webright ) & \text{if $x=\webleft [\webleft (0,a\webright )\webright ]$},\\ \psi \webleft (b\webright ) & \text{if $x=\webleft [\webleft (1,b\webright )\webright ]$} \end{cases} \]for each $x\in A\mathbin {\textstyle \coprod _{C}}B$, which is the unique map making the diagram
commute.
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Associativity. Given a diagram
in $\mathsf{Sets}$, we have isomorphisms of sets
\[ \webleft (A\mathbin {\textstyle \coprod _{X}}B\webright )\mathbin {\textstyle \coprod _{Y}}C\cong \webleft (A\mathbin {\textstyle \coprod _{X}}B\webright )\mathbin {\textstyle \coprod _{B}}\webleft (B\mathbin {\textstyle \coprod _{Y}}C\webright ) \cong A\mathbin {\textstyle \coprod _{X}}\webleft (B\mathbin {\textstyle \coprod _{Y}}C\webright ) \]where these pullbacks are built as in the diagrams
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Interaction With Composition. Given a diagram
in $\mathsf{Sets}$, we have isomorphisms of sets
\begin{align*} X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{\phi \circ f,\psi \circ g}_{K}Y & \cong \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{\phi ,j_{1}}_{A}\webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{f,g}_{K}B\webright )\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{i_{2},i_{1}}_{A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{f,g}_{K}B}\webleft (\webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{f,g}_{K}B\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{j_{2},\psi }_{B}Y\webright )\\ & \cong X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{\phi ,i}_{A}\webleft (\webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{f,g}_{K}B\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{j_{2},\psi }_{B}Y\webright )\\ & \cong \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{\phi ,i_{1}}_{A}\webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{f,g}_{K}B\webright )\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{j,\psi }_{B}Y \end{align*}where
\[ \begin{aligned} j_{1} & = \mathrm{inj}^{A\times ^{f,g}_{K}B}_{1},\\ i_{1} & = \mathrm{inj}^{\webleft (A\times ^{f,g}_{K}B\webright )\times ^{q_{2},\psi }_{Y}}_{1},\\ i & = j_{1}\circ \mathrm{inj}^{\webleft (A\times ^{f,g}_{K}B\webright )\times ^{q_{2},\psi }_{B}Y}_{1}, \end{aligned} \qquad \begin{aligned} j_{2} & = \mathrm{inj}^{A\times ^{f,g}_{K}B}_{2},\\ i_{2} & = \mathrm{inj}^{X\times ^{\phi ,q_{1}}_{A\times ^{f,g}_{K}B}\webleft (A\times ^{f,g}_{K}B\webright )}_{2},\\ j & = j_{2}\circ \mathrm{inj}^{X\times ^{\phi ,q_{1}}_{A}\webleft (A\times ^{f,g}_{K}B\webright )}_{2}, \end{aligned} \]and where these pullbacks are built as in the diagrams
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Unitality. We have isomorphisms of sets natural in $\webleft (A,f\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{X/}\webright )$.
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Commutativity. We have an isomorphism of sets natural in $\webleft (A,f\webright ),\webleft (B,g\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{X/}\webright )$.
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Interaction With Coproducts. We have
- Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{X/},\mathbin {\textstyle \coprod _{X}},X\webright )$ is a symmetric monoidal category.
