Two morphisms of category $\ \mathbf C\ $ are isomorphic to one another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that has the vertical edges (aimed North) being isomorphisms of $\ \mathbf C$.
Problem What is the minimum total number of morphisms of a category such that there are isomorphic morphisms $\ f\ $ and $\ u,\ $ and another isomorphic pair $\ g\ $ and $\ v,\ $ and the compositions $\ g\circ f\ $ and $\ v\circ u\ $ exist but are not isomorphic?
I have an example of a category, as described above, that has a total number of $27$ morphisms. (No, I've touched NO computer :) ).
AN ADDITIONAL NOTE:
Now, that @HenrikRüping has provided his excellent example (most likely minimal), let me mention that my example was a monoid too (but of course) of all maps into itself of a 3-element set.