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.
-
1.
Characterisations. The following conditions are equivalent:
-
(a)
The functor $F$ is a monomorphism of categories.
-
(b)
The functor $F$ is injective on objects and morphisms, i.e. $F$ is injective on objects and the map
\[ F\colon \operatorname {\mathrm{Mor}}\webleft (\mathcal{C}\webright )\to \operatorname {\mathrm{Mor}}\webleft (\mathcal{D}\webright ) \]is injective.
-
(a)
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:
-
1.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
\[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]is a monomorphism of categories?
-
2.
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 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].
