Let $X$ be a set.
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Functoriality I. The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines a functor
\[ \mathcal{P}_{!}\colon \mathsf{Sets}\to \mathsf{Sets}, \]where
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Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, we have
\[ \mathcal{P}_{!}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ). \] -
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Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, 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 4.6.1.1.1.
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2.
Functoriality II. The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines a functor
\[ \mathcal{P}^{-1}\colon \mathsf{Sets}^{\mathsf{op}}\to \mathsf{Sets}, \]where
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Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, we have
\[ \mathcal{P}^{-1}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ). \] -
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Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, the action on morphisms
\[ \mathcal{P}^{-1}_{A,B}\colon \mathsf{Sets}\webleft (A,B\webright )\to \mathsf{Sets}\webleft (\mathcal{P}\webleft (B\webright ),\mathcal{P}\webleft (A\webright )\webright ) \]of $\mathcal{P}^{-1}$ at $\webleft (A,B\webright )$ is the map defined by by sending a map of sets $f\colon A\to B$ to the map
\[ \mathcal{P}^{-1}\webleft (f\webright )\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]defined by
\[ \mathcal{P}^{-1}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f^{-1}, \]as in Definition 4.6.2.1.1.
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3.
Functoriality III. The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines a functor
\[ \mathcal{P}_{*}\colon \mathsf{Sets}\to \mathsf{Sets}, \]where
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Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, we have
\[ \mathcal{P}_{*}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ). \] -
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Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, 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 4.6.3.1.1.
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Proof of Proposition 4.4.2.1.1.
Item 1: Functoriality I
This follows from Item 3 and Item 4 of Proposition 4.6.1.1.6.
Item 2: Functoriality II
This follows from Item 3 and Item 4 of Proposition 4.6.2.1.4.
Item 3: Functoriality III
This follows from Item 3 and Item 4 of Proposition 4.6.3.1.8.
