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