How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right? But remember, I want to understand isomorphisms as "corresponding to" natural isomorphisms between $\mathrm{Hom}(X,−)$ and $\mathrm{Hom}(Y,−)$. I want to not have to choose them. So suppose $\mathbf{C}$ is an $\mathcal{M}$-category. Then, from the perspective of hom-functors, how can we justify the idea that an "isomorphism" $X \rightarrow Y$ is the same as a "tight isomorphism" $X \rightarrow Y$?