The Clowder Project

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

1. 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 $?

2. 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].