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](https://ncatlab.org/nlab/show/locally+cartesian+closed+category).


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](https://ncatlab.org/nlab/show/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](https://ncatlab.org/nlab/show/parametric+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](https://ncatlab.org/nlab/show/locally+fully+faithful+2-functor), [local adjunction](https://ncatlab.org/nlab/show/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:


1. 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,
2. 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:


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


Unfortunately, it turns out that [locally fully faithful](https://ncatlab.org/nlab/show/locally+fully+faithful+2-functor) 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.



Overloading of the word "local" in category theory
The terminology situation is certainly unfortunate. There is an nLab page for [locally](https://ncatlab.org/nlab/show/locally), which lists the different usages of the term (it doesn't currently include those for "local" rather than "locally"). The page suggests "slice-wise" to disambiguate the first case, and "hom-wise" to disambiguate the second case.


