The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the following ways, which AFAIU are quite unrelated:

A category is locally $P$ if all of its slice categories are $P$.

Example: locally cartesian closed categories.

This is applied to functors in two ways:

A functor $F : \mathcal C \to \mathcal D$ is locally $P$ if its restriction to slice categories $F/x : \mathcal C / x \to \mathcal D / Fx$ is $P$.

Example: locally cartesian closed functor.

A functor $F : \mathcal C \to \mathcal D$ is locally $P$ if its restriction $F/\top : \mathcal C \to \mathcal D / F \top$ (where $\top$ is the terminal object) is $P$.

Example: local right adjoint functor.

For (2-)categories and (2-)functors, locally $P$ means that $P$ holds on Hom-sets/categories.

Examples: locally small, locally fully faithful, local adjunction, ...

Locally $P$ means that $P$ holds after localization/sheafification (or similar).

Examples: localization, local epimorphism, local object, local equivalence, ...

Am I correct that these uses are fairly unrelated? If so, are there synonyms that I can use to disambiguate? The current situation makes it difficult to come up with good terminology for things. For example, I am looking for names for the following concepts:

- A functor $F : \mathcal C \to \mathcal D$ whose restriction $F/\top : \mathcal C \to \mathcal D / F \top$ is (a) full / (b) faithful / (c) fully faithful,
- A functor $F : \mathcal C \to \mathcal D$ whose restrictions to slice categories $F/x : \mathcal C / x \to \mathcal D / Fx$ are all (a) full / (b) faithful / (c) fully faithful.

I don't care that 1b is equivalent to 2b and 1c to 2c. What I'm looking for is a general way to come up with good names for concepts. Currently, the best names I can come up with are:

- (a) locally full / (b) locally faithful / (c) locally fully faithful,
- (a) locally full / (b) locally faithful / (c) locally fully faithful, (yes, that's the same).

Unfortunately, it turns out that locally fully faithful is already the term used for 2-functors that are fully faithful on Hom-categories. So I have 6 concepts, for which I can come up with only 3 terms which moreover are already taken.