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Good point, the literature on 2-categories is pretty bad. I just checked the references that we have at 2-category, and none of them admits to a definition of “fully faithful” 2-functors (which does not stop some from using the term anyway). The closest is Section 7 in
which I recommend as the primary source for reading up on the subject. These authors don’t quite say “fully faithful as a 2-functor” but they do equivalently say (Def. 7.0.1) “essentially full on 1-cells and fully faithful on 2-cells” which is, equivalently, the expected “equivalence on all hom-categories” (by the classical theorem here).
But no matter the terminology, the definition is certainly correct and universally accepted: The point is the theorem (7.4.1 in the above book, I have now made a note on that here) – which holds in great generality throuhghout higher category theory, that (assuming the axiom of choice) a morphism of any type of higher categories is an equivalence precisely if, first, it is surjective on equivalence class of objects, and, second, is fully faithful in the sense that for any pair of objects its component on hom-objects is an equivalence of hom-objects, in whatever sense of higher categories these hom-objects themselves are.
From this theorem the respective notion of full higher sub-category is implied: It must be an equivalence onto its image, hence it must satisfy the above fully faithfulness-condition, while it may violate the essential surjectivity on objects, and of course will so to the extent that it is a proper (full, higher) sub-category.
Ah, these authors here say fully explicitly what you are asking for:
See: Def. 2.4.9.
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