A functor $F\colon \mathcal{C}\to \mathcal{D}$ is a monomorphism of categories if it is a monomorphism in $\mathsf{Cats}$ (see 
, ).
11.7.2 Monomorphisms of Categories
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
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1.
Characterisations. The following conditions are equivalent:
Proof of Proposition 11.7.2.1.2.
Item 1: Characterisations
Omitted.
Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:
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1.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor
\[ F^{*}\colon \mathsf{Fun}(\mathcal{D},\mathcal{X})\to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]is a monomorphism of categories?
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2.
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 a monomorphism of categories?
This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
