The Clowder Project

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

1. Functoriality. The assignments $\mathcal{C},\mathcal{D},\webleft (\mathcal{C},\mathcal{D}\webright )\mapsto \mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ define functors
   \[ \begin{array}{ccc} \mathsf{Fun}\webleft (\mathcal{C},-\webright )\colon \mkern -15mu & \mathsf{Cats} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats},\\ \mathsf{Fun}\webleft (-,\mathcal{D}\webright )\colon \mkern -15mu & \mathsf{Cats}\mathrlap {{}^{\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats},\\ \mathsf{Fun}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Cats}^{\mathsf{op}}\times \mathsf{Cats} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}. \end{array} \]
2. 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
   \[ \begin{array}{ccc} \mathsf{Fun}\webleft (\mathcal{C},-\webright )\colon \mkern -15mu & \mathsf{Cats}_{\mathsf{2}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}_{\mathsf{2}},\\ \mathsf{Fun}\webleft (-,\mathcal{D}\webright )\colon \mkern -15mu & \mathsf{Cats}_{\mathsf{2}}^{\mathrlap {\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}_{\mathsf{2}},\\ \mathsf{Fun}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Cats}_{\mathsf{2}}^{\mathsf{op}}\times \mathsf{Cats}_{\mathsf{2}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Cats}_{\mathsf{2}}. \end{array} \]
3. 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 $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.

4. 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 $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}_{\mathsf{2}}\webright )$.

5. Interaction With Punctual Categories. We have a canonical isomorphism of categories
   \[ \mathsf{Fun}\webleft (\mathsf{pt},\mathcal{C}\webright ) \cong \mathcal{C}, \]

   natural in $\mathcal{C}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.

6. 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 $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.

7. Interaction With Co/Completeness. If $\mathcal{E}$ is co/complete, then so is $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )$.
8. 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:
   1. The natural transformation
      \[ \alpha \colon F \Longrightarrow G \]

      is a monomorphism (resp. epimorphism) in $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.

   2. 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 $\mathcal{D}$.