Action on Morphisms. For each $\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\in \text{Obj}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, the action on $\textup{Hom}$-sets
\[ \webleft(\prod _{i\in I}\webright)_{\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}} \colon \text{Nat}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )\to \mathsf{Sets}\webleft(\prod _{i\in I}A_{i},\prod _{i\in I}B_{i}\webright) \]
of $\prod _{i\in I}$ at $\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ is defined by sending a map
\[ \webleft\{ f_{i}\colon A_{i}\to B_{i} \webright\} _{i\in I} \]
in $\text{Nat}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ to the map of sets
\[ \prod _{i\in I}f_{i}\colon \prod _{i\in I}A_{i}\to \prod _{i\in I}B_{i} \]
defined by
\[ \webleft[\prod _{i\in I}f_{i}\webright]\webleft (\webleft (a_{i}\webright )_{i\in I}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{i}\webleft (a_{i}\webright )\webright )_{i\in I} \]
for each $\webleft (a_{i}\webright )_{i\in I}\in \prod _{i\in I}A_{i}$.