Distributivity Over Coproducts. Let $A$, $B$, and $C$ be sets and let $\phi _{A}\colon A\to X$, $\phi _{B}\colon B\to X$, and $\phi _{C}\colon C\to X$ be morphisms of sets. We have isomorphisms of sets
\begin{align*} \delta ^{\mathsf{Sets}_{/X}}_{\ell } & \colon A\times _{X}\webleft (B\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}C\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (A\times _{X}B\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (A\times _{X}C\webright ),\\ \delta ^{\mathsf{Sets}_{/X}}_{r} & \colon \webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\webright )\times _{X}C \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (A\times _{X}C\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (B\times _{X}C\webright ), \end{align*}
as in the diagrams
natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}_{/X}\webright )$.