Full and faithful functors

In category theory, a faithful functor (respectively a full functor) is a functor that is injective (respectively surjective) when restricted to each set of morphisms that have a given source and target.
Formal definitions.
Explicitly, let "C" and "D" be (locally small) categories and let "F" : "C" → "D" be a functor from "C" to "D". The functor "F" induces a function
for every pair of objects "X" and "Y" in "C". The functor "F" is said to be
for each "X" and "Y" in "C".
Properties.
A faithful functor need not be injective on objects or morphisms. That is, two objects "X" and "X"′ may map to the same object in "D" (which is why the range of a full and faithful functor is not necessarily isomorphic to "C"), and two morphisms "f" : "X" → "Y" and "f"′ : "X"′ → "Y"′ (with different domains/codomains) may map to the same morphism in "D". Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in "D" not of the form "FX" for some "X" in "C". Morphisms between such objects clearly cannot come from morphisms in "C".
A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if "F" : "C" → "D" is a full and faithful functor and formula_2 then formula_3.
Generalization to (∞, 1)-categories.
The notion of a functor being 'full' or 'faithful' does not translate to the notion of a (∞, 1)-category. In an (∞, 1)-category, the maps between any two objects are given by a space only up to homotopy. Since the notion of injection and surjection are not homotopy invariant notions (consider an interval embedding into the real numbers vs. an interval mapping to a point), we do not have the notion of a functor being "full" or "faithful." However, we can define a functor of quasi-categories to be "fully faithful" if for every "X" and "Y" in "C," the map formula_4is a weak equivalence.