Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
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Functoriality. The assignments $\mathcal{C},\mathcal{D},\webleft (\mathcal{C},\mathcal{D}\webright )\mapsto \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ define functors
\begin{gather*} \begin{aligned} \mathsf{Fun}\webleft (\mathcal{C},-_{2}\webright ) & \colon \mathsf{Cats}\to \mathsf{Cats},\\ \mathsf{Fun}\webleft (-_{1},\mathcal{D}\webright ) & \colon \mathsf{Cats}^{\mathsf{op}} \to \mathsf{Cats}, \end{aligned}\\ \mathsf{Fun}\webleft (-_{1},-_{2}\webright ) \colon \mathsf{Cats}^{\mathsf{op}}\times \mathsf{Cats}\to \mathsf{Cats}. \end{gather*}
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2-Functoriality. The assignments $\mathcal{C},\mathcal{D},\webleft (\mathcal{C},\mathcal{D}\webright )\mapsto \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ define 2-functors
\begin{gather*} \begin{aligned} \mathsf{Fun}\webleft (\mathcal{C},-_{2}\webright ) & \colon \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Cats}_{\mathsf{2}},\\ \mathsf{Fun}\webleft (-_{1},\mathcal{D}\webright ) & \colon \mathsf{Cats}_{\mathsf{2}}^{\mathsf{op}} \to \mathsf{Cats}_{\mathsf{2}}, \end{aligned}\\ \mathsf{Fun}\webleft (-_{1},-_{2}\webright ) \colon \mathsf{Cats}_{\mathsf{2}}^{\mathsf{op}}\times \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Cats}_{\mathsf{2}}. \end{gather*}
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Adjointness. We have adjunctions witnessed by bijections of sets
\begin{align*} \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{D},\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )\webright ),\\ \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathsf{Fun}\webleft (\mathcal{D},\mathcal{E}\webright )\webright ), \end{align*}
natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.
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2-Adjointness. We have 2-adjunctions witnessed by isomorphisms of categories
\begin{align*} \mathsf{Fun}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \mathsf{Fun}\webleft (\mathcal{D},\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )\webright ),\\ \mathsf{Fun}\webleft (\mathcal{C}\times \mathcal{D},\mathcal{E}\webright ) & \cong \mathsf{Fun}\webleft (\mathcal{C},\mathsf{Fun}\webleft (\mathcal{D},\mathcal{E}\webright )\webright ), \end{align*}
natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}_{\mathsf{2}}\webright )$.
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Interaction With Punctual Categories. We have a canonical isomorphism of categories
\[ \mathsf{Fun}\webleft (\mathsf{pt},\mathcal{C}\webright ) \cong \mathcal{C}, \]
natural in $\mathcal{C}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.
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Objectwise Computation of Co/Limits. Let
\[ D \colon \mathcal{I} \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright ) \]
be a diagram in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$. We have isomorphisms
\begin{align*} \operatorname*{\text{lim}}\webleft (D\webright )_{A} & \cong \operatorname*{\text{lim}}_{i\in \mathcal{I}}\webleft (D_{i}\webleft (A\webright )\webright ),\\ \operatorname*{\text{colim}}\webleft (D\webright )_{A} & \cong \operatorname*{\text{colim}}_{i\in \mathcal{I}}\webleft (D_{i}\webleft (A\webright )\webright ), \end{align*}
naturally in $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.
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Interaction With Co/Completeness. If $\mathcal{E}$ is co/complete, then so is $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )$.
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Monomorphisms and Epimorphisms. Let $\alpha \colon F\Longrightarrow G$ be a morphism of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$. The following conditions are equivalent:
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The natural transformation
\[ \alpha \colon F \Longrightarrow G \]
is a monomorphism (resp. epimorphism) in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
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For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the morphism
\[ \alpha _{A} \colon F_{A} \to G_{A} \]
is a monomorphism (resp. epimorphism) in $\mathcal{D}$.