The Clowder Project

Let $\webleft\{ A_{i}\webright\} _{i\in I}$ be a family of sets.

1. Functoriality. The assignment $\webleft\{ A_{i}\webright\} _{i\in I}\mapsto \prod _{i\in I}A_{i}$ defines a functor
   \[ \prod _{i\in I}\colon \mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\to \mathsf{Sets} \]

   where

   • Action on Objects. For each $\webleft (A_{i}\webright )_{i\in I}\in \text{Obj}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, we have
     \[ \webleft[\prod _{i\in I}\webright]\webleft (\webleft (A_{i}\webright )_{i\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\prod _{i\in I}A_{i} \]

   • 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 $\webleft (a_{i}\webright )_{i\in I}\in \prod _{i\in I}A_{i}$.