
Special Property?
Is there any name given to this: Suppose $\displaystyle \langle S, * \rangle $, $\displaystyle \langle S', *' \rangle $ and $\displaystyle \langle S'', *'' \rangle $ are binary structures. Suppose there is an isomorphism $\displaystyle \phi $ between $\displaystyle S $ and $\displaystyle S' $. Also suppose $\displaystyle S $ and $\displaystyle S' $ are isomorphic. Also $\displaystyle S' $ and $\displaystyle S'' $ are isomorphic, and the same isomorphism $\displaystyle \phi $ are used.

By definition, an isomorphism is a function.
Two functions are identical only if the domains and ranges are the same.
So unless $\displaystyle S = S' = S''$ it's not possible for the same isomorphism to be used for $\displaystyle \phi: S \to S'$ as it is for $\displaystyle \phi: S' \to S''$ 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.