The Clowder Project

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

1. 1. For each $\mathcal{X}\in \text{Obj}\left(\mathsf{Cats}\right)$, the precomposition functor
   $$F^{\ast }:\mathsf{Fun}(\mathcal{D},\mathcal{X})\to \mathsf{Fun}(\mathcal{C},\mathcal{X})$$

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

2. 2. For each $\mathcal{X}\in \text{Obj}\left(\mathsf{Cats}\right)$, the postcomposition functor
   $$F_{\ast }:\mathsf{Fun}(\mathcal{X},\mathcal{C})\to \mathsf{Fun}(\mathcal{X},\mathcal{D})$$

   is essentially injective, i.e. if $F\circ \varphi \cong F\circ \psi$, then $\varphi \cong \psi$?

This question also appears as [MO 468125].