Is there any name given to this: Suppose , and are binary structures. Suppose there is an isomorphism between and . Also suppose and are isomorphic. Also and are isomorphic, and the same isomorphism are used.
By definition, an isomorphism is a function.
Two functions are identical only if the domains and ranges are the same.
So unless it's not possible for the same isomorphism to be used for as it is for because by definition the two isomorphism would need to be different.
As to whether there's a name for this or not, I haven't a clue, but unless there are some unspoken assumptions (or it's a trick question), I'd question what's going on here.