The Clowder Project

where

• Action on Objects. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
  \[ \text{Gr}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A. \]

• Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on $\textup{Hom}$-sets
  \[ \text{Gr}_{A,B}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \underbrace{\mathrm{Rel}\webleft (\text{Gr}\webleft (A\webright ),\text{Gr}\webleft (B\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{Rel}\webleft (A,B\webright )} \]

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

  \[ \text{Gr}_{A,B}\webleft (f\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Gr}\webleft (f\webright ), \]

  where $\text{Gr}\webleft (f\webright )$ is the graph of $f$ as in Definition 6.3.1.1.1.

• Preservation of Identities. We have
  \[ \text{Gr}\webleft (\text{id}_{A}\webright )=\chi _{A} \]

  for each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.

• Preservation of Composition. We have
  \[ \text{Gr}\webleft (g\circ f\webright )=\text{Gr}\webleft (g\webright )\mathbin {\diamond }\text{Gr}\webleft (f\webright ) \]

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