Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
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Functionality I. The assignment $R\mapsto R^{-1}$ defines a function
\[ \webleft (-\webright )^{-1}\colon \mathrm{Rel}\webleft (A,B\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ). \]
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Functionality II. The assignment $R\mapsto R^{-1}$ defines a function
\[ \webleft (-\webright )^{-1}\colon \mathrm{Rel}\webleft (A,B\webright ) \to \mathsf{Pos}\webleft (\webleft (\mathcal{P}\webleft (A\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )\webright ). \]
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Interaction With Identities. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have[1]
\[ \webleft (\chi _{A}\webright )^{-1}=\text{id}_{\mathcal{P}\webleft (A\webright )}. \]
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Interaction With Composition. For each pair of composable relations $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ and $S\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}C$, we have[2]
