The Clowder Project

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

1. 1.

   Characterisations. If $\mathcal{C}$ and $\mathcal{D}$ are small, then the following conditions are equivalent:

   1. (a)

      The functor $F$ is an isomorphism of categories.

   2. (b)

      The functor $F$ is fully faithful and bijective on objects.

   3. (c)

      For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, 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. (d)

      For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, 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.