The Clowder Project

where ${-}_1\coprod {-}_2$ is the functor where

• Action on Objects. For each $(A,B)\in \text{Obj}(\mathsf{Sets}\times \mathsf{Sets})$, we have
  $$\left[{-}_1\coprod {-}_2\right](A,B)\stackrel{\text{def}}{=}A\coprod B.$$

• Action on Morphisms. For each $(A,B),(X,Y)\in \text{Obj}\left(\mathsf{Sets}\right)$, the action on $\text{Hom}$-sets
  $$\coprod _{(A,B),(X,Y)}:\mathsf{Sets}(A,X)\times \mathsf{Sets}(B,Y)\to \mathsf{Sets}(A\coprod B,X\coprod Y)$$

  of $\coprod$ at $((A,B),(X,Y))$ is defined by sending $(f,g)$ to the function

  $$f\coprod g:A\coprod B\to X\coprod Y$$

  defined by

  $$\left[f\coprod g\right]\left(x\right)\stackrel{\text{def}}{=}\{\begin{array}{ll}(0,f\left(a\right))& \text{if }x=(0,a)\text{,}\\ (1,g\left(b\right))& \text{if }x=(1,b)\text{,}\end{array}$$

  for each $x\in A\coprod B$.