We denote natural transformations in diagrams as
-
1.
Interaction With Postcomposition. The following conditions are equivalent:
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(a)
The functor $F\colon \mathcal{C}\to \mathcal{D}$ is full.
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(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.
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(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 14: Types of Morphisms in Bicategories, Definition 14.1.2.1.1.
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(a)
15.2.4 Categories
(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.)
