# Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc

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:

1. 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).
2. [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:

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

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

1. $$F_*$$ is conservative.
2. $$F_*$$ is essentially injective, i.e. $$F\circ\phi\cong F\circ\psi$$ implies $$\phi\cong\psi$$.
3. $$F^*$$ is essentially injective, i.e. $$\phi\circ F\cong\psi\circ F$$ implies $$\phi\cong\psi$$.
4. $$F_*$$ (resp. $$F^*$$) is essentially surjective.
5. $$F_*$$ (resp. $$F^*$$) is a monomorphism (resp. an epimorphism).
6. $$F_*$$ (resp. $$F^*$$) is pseudomonic (resp. pseudoepic).
7. $$F_*$$ (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.