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 $F:D\to C$ 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 $C$.