A subcategory $\mathcal{A}$ of a category $\mathcal{B}$ is said to be isomorphism-closed if for any $A\in\mathcal{A}$ and a $\mathcal{B}$-isomorphism $h:A\to B$, also the $\mathcal{B}$-object $B$ belongs to $\mathcal{A}$.

More simply: the subcategory $\mathcal{A}$ contains with each object all isomorphic $\mathcal{B}$-objects.

Another name commonly used for isomorphism-closed subcategories is replete subcategory.

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 $\mathbf{Top}$.

A subcategory which is isomorphism-closed and full is called strictly full.