The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines functors1
\begin{align*} \mathcal{P}_{*} & \colon \mathrm{Rel}\to \mathsf{Sets},\\ \mathcal{P}_{-1} & \colon \mathrm{Rel}^{\mathsf{op}} \to \mathsf{Sets},\\ \mathcal{P}^{-1} & \colon \mathrm{Rel}^{\mathsf{op}} \to \mathsf{Sets},\\ \mathcal{P}_{!} & \colon \mathrm{Rel}\to \mathsf{Sets}\end{align*}
where
- Action on Objects. For each $A\in \text{Obj}\webleft (\mathrm{Rel}\webright )$, we have
\begin{align*} \mathcal{P}_{*}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ),\\ \mathcal{P}_{-1}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ),\\ \mathcal{P}^{-1}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ),\\ \mathcal{P}_{!}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright );\end{align*}
- Action on Morphisms. For each morphism $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ of $\mathrm{Rel}$, the images
\begin{align*} \mathcal{P}_{*}\webleft (R\webright ) & \colon \mathcal{P}\webleft (A\webright ) \to \mathcal{P}\webleft (B\webright ),\\ \mathcal{P}_{-1}\webleft (R\webright ) & \colon \mathcal{P}\webleft (B\webright ) \to \mathcal{P}\webleft (A\webright ),\\ \mathcal{P}^{-1}\webleft (R\webright ) & \colon \mathcal{P}\webleft (B\webright ) \to \mathcal{P}\webleft (A\webright ),\\ \mathcal{P}_{!}\webleft (R\webright ) & \colon \mathcal{P}\webleft (A\webright ) \to \mathcal{P}\webleft (B\webright )\end{align*}
of $R$ by $\mathcal{P}_{*}$, $\mathcal{P}_{-1}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{!}$ are defined by
\begin{align*} \mathcal{P}_{*}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{*},\\ \mathcal{P}_{-1}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1},\\ \mathcal{P}^{-1}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{-1},\\ \mathcal{P}_{!}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{!},\end{align*}as in Definition 7.4.1.1.1, Definition 7.4.2.1.1, Definition 7.4.3.1.1, and Definition 7.4.4.1.1.
1The functor $\mathcal{P}_{*}\colon \mathrm{Rel}\to \mathsf{Sets}$ admits a left adjoint; see Item 3 of Proposition 7.3.1.1.2.
