nLab
full subcategory

A subcategory S of a category C is full if its inclusion functor? is full; this inclusion functor is often called a full embedding or a full inclusion.

To specify a full subcategory S of C, it is enough to say which objects belong to S. Then S must consist of all morphisms whose source and target belong to S (and no others). One speaks of the full subcategory on a given set of objects.

Any full and faithful functor F:DC defines a full subcategory of C, the full subcategory on the image (or essential image) of F, which (either way) will be equivalent to D. Thus we may consider any category D equipped with such a functor to be a full subcategory of C.