Let $f\colon X\to B$ be a function.
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Functionality I. The assignment $f\mapsto f_{!}$ defines a function
\[ \webleft (-\webright )_{!|X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (X\webright ),\mathcal{P}\webleft (Y\webright )\webright ). \]
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Functionality II. The assignment $f\mapsto f_{!}$ defines a function
\[ \webleft (-\webright )_{!|X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Pos}\webleft (\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright )\webright ). \]
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Interaction With Identities. For each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
\[ \webleft (\text{id}_{X}\webright )_{!}=\text{id}_{\mathcal{P}\webleft (X\webright )}. \]
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Interaction With Composition. For each pair of composable functions $f\colon X\to Y$ and $g\colon Y\to Z$, we have
