The Clowder Project

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

1. 1.

   For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the precomposition functor

   \[ F^{*}\colon \mathsf{Fun}(\mathcal{D},\mathcal{X})\to \mathsf{Fun}(\mathcal{C},\mathcal{X}) \]

   is essentially surjective?

2. 2.

   For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the postcomposition functor

   \[ F_{*}\colon \mathsf{Fun}(\mathcal{X},\mathcal{C})\to \mathsf{Fun}(\mathcal{X},\mathcal{D}) \]

   is essentially surjective?

This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].