I'm currently writing a comprehensive/encyclopedic set of notes on category theory, and one of the things I'm trying to do is gather all sorts of statements of the form

Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. The following conditions are equivalent:

- For each $X\in\mathrm{Obj}(\mathsf{Cats})$, the precomposition functor $$F^* \colon \mathsf{Fun}(\mathcal{D},\mathcal{X}) \to \mathsf{Fun}(\mathcal{C},\mathcal{X})$$ is X for X some nice property (e.g. conservative).
- [Other conditions on $F$ itself]

or with postcomposition $$F_*\colon\mathsf{Fun}(\mathcal{X},\mathcal{C})\to\mathsf{Fun}(\mathcal{X},\mathcal{D})$$ replacing $F^*$.

So far, I've found characterisations for some nice properties, but there are many, many ones I still haven't found anything about.

Thus, I thought it might be a good idea to collect those into a single question here on MO, to keep track of all those:

Known characterisations (I'll link them here once the notes are ready):

- $F_*$ is full/faithful/fully faithful.
- $F^*$ is full/faithful/fully faithful (though see also MO 468121).
- $F^*$ is conservative.

Characterisations which I haven't yet found a reference for or worked out:

- $F_*$ is conservative.
- $F_*$ is essentially injective, i.e. $F\circ\phi\cong F\circ\psi$ implies $\phi\cong\psi$.
- $F^*$ is essentially injective, i.e. $\phi\circ F\cong\psi\circ F$ implies $\phi\cong\psi$.
- $F_*$ (resp. $F^*$) is essentially surjective.
- $F_*$ (resp. $F^*$) is a monomorphism (resp. an epimorphism).
- $F_*$ (resp. $F^*$) is pseudomonic (resp. pseudoepic).
- $F_*$ (resp. $F^*$) is dominant.

I'm also interested on the following two natural refinements of Items 2 and 3 above:

- The functor $$F_*\colon\mathsf{Core}(\mathsf{Fun}(\mathcal{X},\mathcal{C}))\to\mathsf{Core}(\mathsf{Fun}(\mathcal{X},\mathcal{D}))$$ is full, i.e. if $F\circ\phi\cong F\circ\psi$ via a natural isomorphism $\beta$, then $\phi\cong\psi$ via a natural isomorphism $\alpha$ and we have $\beta=\mathrm{id}_{F}\star\alpha$.
- The functor $$F^*\colon\mathsf{Core}(\mathsf{Fun}(\mathcal{D},\mathcal{X}))\to\mathsf{Core}(\mathsf{Fun}(\mathcal{C},\mathcal{X}))$$ is full, i.e. if $\phi\circ F\cong \psi\circ F$ via $\beta$, then $\phi\cong\psi$ via $\alpha$ and $\beta=\alpha\star\mathrm{id}_{F}$.

A few of these seem hard enough that probably no answer is currently known (or way too hard, like epimorphisms), but I suspect some of these are things that have been worked out already somewhere or are possibly folklore among the correct experts, but so far I haven't had much luck procuring references or statements for them.