Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute? 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.
 A: 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 $A$.
A: Here is an example. Consider the family of rotated copies of the unit square in the plane $\mathbb{R}\times\mathbb{R}$, rotated by some angle about the origin. If one such copy is rotated only a little from a second, then it seems the canonical isometry to bring them into alignment would be to rotate the first through the smallest possible angle to bring it into alignment with the second. But of course, the composition of many such small angle rotations  (or even just two of sufficient small size) would add up to a large angle, and so the composition of these canonical isometries is no longer canonical.
(I suppose that one should say here that if the figures are rotated by exactly $45^\circ$, then either of the two minimal rotations should count as canonical.)
A: Let $X=\mathbb{A}_{\mathbb{C}}^1\backslash\{0\}$. The first algebraic de Rham cohomology of $X$ is 1-dimensional, and has a canonical generator $\frac{dz}{z}$ (or any other 1-form with residue 1). The first Betti cohomology is also 1-dimensional, and has a canonical generator taking a 1-cycle to its winding number around 0. Over $\mathbb{C}$ there is a canonical isomorphism $H^1_{dR}(X)\to H^1_B(X,\mathbb{C})$, but it takes the generator to $2\pi i$ times the generator.
If we wanted to say this in terms of isomorphisms, we could consider the three vector spaces $\mathbb{C},H^1_{dR}(X)$, and $H^1(X,\mathbb{C})$, and use the fact that an isomorphism from $\mathbb{C}$ to a 1-dimensional vector space is the same thing as the choice of a generator of that vector space.
A: You don't really need to construct sophisticated mathematical examples to answer this. The problem is that the word canonical has at best a sociological meaning, not a mathematical one.
Are $X\times Y$ and $Y\times X$ "canonically isomorphic" by the switching map?
In some contexts it may be reasonable to say so, in order to avoid bureaucratic notation.  However, once $X$ is non-trivial and $Y=X$ you have a non-trivial group.
All that word canonical means is "I'm too lazy to write down the definition".
