The Clowder Project

²

1. The functor $F$ is an equivalence of categories.
2. The functor $F$ is fully faithful and essentially surjective.
3. The induced functor
   \[ \left.F\right\vert _{\mathsf{Sk}\webleft (\mathcal{C}\webright )}\colon \mathsf{Sk}\webleft (\mathcal{C}\webright )\to \mathsf{Sk}\webleft (\mathcal{D}\webright ) \]

   is an isomorphism of categories.

4. For each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
   \[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

   is an equivalence of categories.

5. For each $X\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
   \[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

   is an equivalence of categories.