In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target.

In other words, a functor F : C → D is full if the maps

F_X,Y : Mor_C(X, Y) → Mor_D(FX, FY)

are surjective for every pair of objects X and Y in C.

Note that a full functor need not be surjective on objects or morphisms. That is, 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.

For example, let F : C → Set be the functor which maps every object in C to the empty set and every morphism to the empty function. Then F is full, but neither surjective on objects or morphisms.

Another example is the forgetful functor Ab → Grp. This is full, but neither surjective on objects or morphisms. A counterexample is the forgetful functor Grp → Set. This is not full as there are functions between group which are not group homomorphisms.

See also:

• faithful functor
• full subcategory