If one accepts the notion of subcategory without any qualification (as discussed there), then:

A subcategory$S$ of a category $C$ is a full subcategory if for any $x$ and $y$ in $S$, every morphism $f : x \to y$ in $C$ is also in $S$ (that is, the inclusion functor$S \hookrightarrow C$ is full).

This inclusion functor is often called a full embedding or a full inclusion.

Notice that 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.