The Clowder Project

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

1. Interaction With Natural Isomorphisms. The following conditions are equivalent:
   1. The natural transformation $F^{\dagger }\colon \textup{Hom}_{\mathcal{C}}\Longrightarrow {\textup{Hom}_{\mathcal{D}}}\circ {\webleft (F^{\mathsf{op}}\times F\webright )}$ associated to $F$ is a natural isomorphism.
   2. The functor $F$ is fully faithful.
2. Interaction With Composition. We have an equality of pasting diagrams
   
   in $\mathsf{Cats}_{\mathsf{2}}$, i.e. we have
   \[ \webleft (G\circ F\webright )^{\dagger }=\webleft (G^{\dagger }\mathbin {\star }\text{id}_{F^{\mathsf{op}}\times F}\webright )\circ F^{\dagger }. \]
3. Interaction With Identities. We have
   \[ \text{id}^{\dagger }_{\mathcal{C}}=\text{id}_{\textup{Hom}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )}, \]

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