The Clowder Project

We denote natural transformations in diagrams as

(This tag has been removed and is now part of Chapter 9: Categories, Remark 9.9.2.1.2.)

(This Tag was an item of Chapter 9: Categories, Proposition 9.6.2.1.2, but has since been removed because its statement is incorrect. Naïm Camille Favier provided a counterexample, and the corrected statements now appear as Chapter 9: Categories, Item 2 and Item 3 of Proposition 9.6.2.1.2.)

1. Interaction With Postcomposition. The following conditions are equivalent:
   1. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is full.
   2. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
      \[ F_{*} \colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright ) \to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

      is full.

   3. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably full morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 11: Types of Morphisms in Bicategories, Definition 11.1.2.1.1.