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
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Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathrm{Rel}\webright )$, 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*} -
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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 8.5.1.1.1, Definition 8.5.2.1.1, Definition 8.5.3.1.1, and Definition 8.5.4.1.1.
- 1The functor $\mathcal{P}_{!}\colon \mathrm{Rel}\to \mathsf{Sets}$ admits a left adjoint; see Item 3 of Proposition 8.2.2.1.2.
