The Clowder Project

(This Tag was an item of Chapter 11: Categories, Proposition 11.6.2.1.2, but has since been removed because its statement is incorrect. Naïm Camille Favier provided a counterexample, and the corrected statements now appear as Chapter 11: Categories, Item 2 and Item 3 of Proposition 11.6.2.1.2.)

1. 1.

   Interaction With Postcomposition. The following conditions are equivalent:

   1. (a)

      The functor $F\colon \mathcal{C}\to \mathcal{D}$ is full.

   2. (b)

      For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor

      \[ F_{*} \colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright ) \to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

      is full.

   3. (c)

      The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably full morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 13: Types of Morphisms in Bicategories, Definition 13.1.2.1.1.