The Clowder Project

where

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

• Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on morphisms
  \[ \mathcal{P}_{!|A,B}\colon \mathsf{Sets}\webleft (A,B\webright )\to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ) \]

  of $\mathcal{P}_{!}$ at $\webleft (A,B\webright )$ is the map defined by by sending a map of sets $f\colon A\to B$ to the map

  \[ \mathcal{P}_{!}\webleft (f\webright )\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]

  defined by

  \[ \mathcal{P}_{!}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}, \]

  as in Definition 2.6.3.1.1.