A subcategory which is isomorphism-closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every -object which is isomorphic to an -object is also an -object.
This condition is very natural. E.g in the category of topological spaces we usually study properties which are invariant under homeomorphisms - so called topological properties. Every topological property corresponds to a strictly full subcategory of