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.



Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute?
Maybe this is kind of contrived, but *any* single object with a canonical automorphism that is not the identity is an example of this. In fact, any other example must in some sense include an example like this, if you consider the composition of $A \to B \to C$ with the inverse of $A \to C$ a "canonical" automorphism of $A$.


