A functor F: C → D is faithful if it is injective on each hom-set (F(f) = F(g) implies f = g), and full if it is surjective on each hom-set (every morphism between F(A) and F(B) in D arises as F(f) for some f in C). A fully faithful functor embeds C as a subcategory of D in a strong sense: it reflects isomorphisms and allows C to be identified with its image in D. Forgetful functors are typically faithful but not full; inclusion functors of full subcategories are fully faithful.
Check the forgetful functor from Ab (abelian groups) to Grp: it is faithful (group homomorphisms between abelian groups are the same in both categories) but not full (not every group homomorphism between two abelian groups is an abelian group homomorphism—actually it is, so check another example). Work out when the inclusion of a subcategory is full.
Your prerequisite on functors established that a functor F: C → D must send objects to objects and morphisms to morphisms, preserving composition and identities. But functors can do this in very different ways — some collapse the structure of C, others faithfully preserve it, and others make C appear richer inside D than it actually is. The notions of full and faithful measure how a functor behaves specifically on *morphisms between pairs of objects*, not on objects themselves.
Think of each hom-set C(A, B) (the set of all morphisms from A to B) as a dataset, and F as a function that sends each morphism f: A → B in C to a morphism F(f): F(A) → F(B) in D. Faithfulness means this map is injective: distinct morphisms in C(A, B) produce distinct morphisms in D(F(A), F(B)). If F(f) = F(g) forces f = g, then F is not "forgetting" the distinction between morphisms. Faithful functors preserve morphism identity — they cannot confuse two different morphisms. The forgetful functor from groups to sets is faithful because group homomorphisms are particular set-functions, and if the same set-function arises from two group homomorphisms, those homomorphisms were the same to begin with.
Fullness is the opposite requirement: F is surjective on each hom-set. Every morphism in D(F(A), F(B)) is the image of some morphism in C(A, B). Fullness means that D does not have "extra" morphisms between the images of A and B that don't come from C. The inclusion of a full subcategory is the prototypical example: if you take a subcategory where you keep all morphisms between chosen objects, that inclusion functor is fully faithful. Contrast with a subcategory that restricts to some morphisms: the inclusion is then faithful but not full.
When F is both full and faithful — fully faithful — it embeds C into D in a strong structural sense. A fully faithful functor reflects isomorphisms: if F(f) is an isomorphism in D, then f was already an isomorphism in C. This means F cannot create or destroy isomorphisms between objects. Two objects A and B in C are isomorphic in C if and only if F(A) and F(B) are isomorphic via a morphism in the image of F. This is how category theorists make precise the idea that "the image of F looks exactly like C." It does not mean F is an equivalence of categories, because F might not be surjective on objects — D may contain objects not of the form F(A). The Yoneda embedding — which you will encounter next — is the canonical example of a fully faithful functor, embedding any category into its presheaf category, and understanding full-faithfulness is essential to grasping what the Yoneda lemma actually says.