Subsection 9.6.6 (013J): Essentially Surjective Functors

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

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is essentially surjective¹ if it satisfies the following condition:

For each $D\in \text{Obj}\webleft (\mathcal{D}\webright )$, there exists some object $A$ of $\mathcal{C}$ such that $F\webleft (A\webright )\cong D$.

¹Further Terminology: Also called an eso functor, where the name “eso” comes from essentially surjective on objects.

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 essentially surjective?
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 essentially surjective?

This question also appears as [MO 468125].

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