Subsection 8.6.2 (014W): Monomorphisms of Categories

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Let $\mathcal{C}$ and $\mathcal{D}$ be categories.

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is a monomorphism of categories if it is a monomorphism in $\mathsf{Cats}$ (see ).

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

Proof of Proposition 8.6.2.1.2.

Item 1: Characterisations

Omitted.

Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:

For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, 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?
For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, 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 [MO 468125].

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