Specifically, are there "nice" (ie, not too obscure or contrived) examples of families of objects, where say for each objects $A,B,C$ in the family, the canonical isomorphism from $A\rightarrow C$ is not the composition of the canonical isomorphisms from $A\rightarrow B$ and $B\rightarrow C$?

To illustrate, here's a non-example: The fundamental groups of a space with different base-points are all canonically isomorphic up to inner automorphisms, so given a space $X$, the fundamental groups $\pi_1(X,x)$ as $x$ varies over $X$, together with the conjugacy classes of these 'canonical isomorphisms', form a category.