Subsection 9.6.5 (013D): Essentially Injective Functors

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

For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, if $F\webleft (A\webright )\cong F\webleft (B\webright )$, then $A\cong B$.

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 injective, i.e. if $\phi \circ F\cong \psi \circ F$, then $\phi \cong \psi $ for all functors $\phi $ and $\psi $?
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 injective, i.e. if $F\circ \phi \cong F\circ \psi $, then $\phi \cong \psi $?

This question also appears as [MO 468125].

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