Question 9.6.4.1.3 (013C): Characterisations of Functors With Conservative Pre/Postcomposition

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Table of Contents
Part 3: Category Theory
Chapter 9: Categories
Section 9.6: Conditions on Functors
Subsection 9.6.4: Conservative Functors
Question 9.6.4.1.3: Characterisations of Functors With Conservative Pre/Postcomposition (cite)

Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ satisfying the following condition:

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 conservative?

This question also appears as [MO 468125].

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