The Clowder Project

Let $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ be functors.

1. 1.

   Interaction With Natural Isomorphisms. The following conditions are equivalent:

   1. (a)

      The natural transformation $F^{\dagger }\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\Longrightarrow {\operatorname {\mathrm{Hom}}_{\mathcal{D}}}\circ {\webleft (F^{\mathsf{op}}\times F\webright )}$ associated to $F$ is a natural isomorphism.

   2. (b)

      The functor $F$ is fully faithful.

2. 2.

   Interaction With Composition. We have an equality of pasting diagrams

   
   in $\mathsf{Cats}_{\mathsf{2}}$, i.e. we have
   \[ \webleft (G\circ F\webright )^{\dagger }=\webleft (G^{\dagger }\mathbin {\star }\operatorname {\mathrm{id}}_{F^{\mathsf{op}}\times F}\webright )\circ F^{\dagger }. \]
3. 3.

   Interaction With Identities. We have

   \[ \operatorname {\mathrm{id}}^{\dagger }_{\mathcal{C}}=\operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )}, \]

   i.e. the natural transformation associated to $\operatorname {\mathrm{id}}_{\mathcal{C}}$ is the identity natural transformation of the functor $\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )$.