The Clowder Project

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

1. Characterisations. If $\mathcal{C}$ and $\mathcal{D}$ are small, then the following conditions are equivalent:
   1. The functor $F$ is an isomorphism of categories.
   2. The functor $F$ is fully faithful and bijective on objects.
   3. 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 isomorphism of categories.

   4. 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 isomorphism of categories.