The Clowder Project

We denote natural transformations in diagrams as

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

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

      is full.