The Clowder Project

where

• Action on Objects. For each $A\in \text{Obj}\left(\mathsf{Sets}\right)$, we have
  $$\text{Gr}\left(A\right)\stackrel{\text{def}}{=}A;$$

• Action on Morphisms. For each $A,B\in \text{Obj}\left(\mathsf{Sets}\right)$, the action on $\text{Hom}$-sets
  $${\text{Gr}}_{A,B}:\mathsf{Sets}(A,B)\to \underset{\stackrel{\text{def}}{=}\mathrm{Rel}(A,B)}{\underbrace{\mathrm{Rel}(\text{Gr}\left(A\right),\text{Gr}\left(B\right))}}$$

  of $\text{Gr}$ at $(A,B)$ is defined by

  $${\text{Gr}}_{A,B}\left(f\right)\stackrel{\text{def}}{=}\text{Gr}\left(f\right),$$

  where $\text{Gr}\left(f\right)$ is the graph of $f$ as in Definition 7.3.1.1.1.

• Preservation of Identities. We have
  $$\text{Gr}\left({\text{id}}_A\right)={\chi }_A$$

  for each $A\in \text{Obj}\left(\mathsf{Sets}\right)$.

• Preservation of Composition. We have
  $$\text{Gr}(g\circ f)=\text{Gr}\left(g\right)\diamond \text{Gr}\left(f\right)$$

  for each pair of functions $f:A\to B$ and $g:B\to C$.