The Clowder Project

Let $\left\{A_i\right\}{}_{i\in I}$ be a family of sets.

1. 1. Functoriality. The assignment $\left\{A_i\right\}{}_{i\in I}\mapsto \prod _{i\in I}{A}_i$ defines a functor
   $$\prod _{i\in I}:\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets})\to \mathsf{Sets}$$

   where

   • Action on Objects. For each $\left(A_i\right){}_{i\in I}\in \text{Obj}\left(\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets})\right)$, we have
     $$\left[\prod _{i\in I}\right]\left(\left(A_i\right){}_{i\in I}\right)\stackrel{\text{def}}{=}\prod _{i\in I}{A}_i$$

   • Action on Morphisms. For each $\left(A_i\right){}_{i\in I},\left(B_i\right){}_{i\in I}\in \text{Obj}\left(\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets})\right)$, the action on $\text{Hom}$-sets
     $$\left(\prod _{i\in I}\right){}_{\left(A_i\right){}_{i\in I},\left(B_i\right){}_{i\in I}}:\text{Nat}(\left(A_i\right){}_{i\in I},\left(B_i\right){}_{i\in I})\to \mathsf{Sets}(\prod _{i\in I}{A}_i,\prod _{i\in I}{B}_i)$$

     of $\prod _{i\in I}$ at $(\left(A_i\right){}_{i\in I},\left(B_i\right){}_{i\in I})$ is defined by sending a map

     $$\{f_i:A_i\to B_i\}{}_{i\in I}$$

     in $\text{Nat}(\left(A_i\right){}_{i\in I},\left(B_i\right){}_{i\in I})$ to the map of sets

     $$\prod _{i\in I}{f}_i:\prod _{i\in I}{A}_i\to \prod _{i\in I}{B}_i$$

     defined by

     $$\left[\prod _{i\in I}{f}_i\right]\left(\left(a_i\right){}_{i\in I}\right)\stackrel{\text{def}}{=}\left(f_i\left(a_i\right)\right){}_{i\in I}$$

     for each $\left(a_i\right){}_{i\in I}\in \prod _{i\in I}{A}_i$.