The classic coherence theorems of MacLane (Natural associativity and commutativity, Rice U. studies, 1963) talked about natural transformations between functors. By 1971 (Kelly-MacLane, Coherence in closed categories, JPAA) and 1972 (Kelly, An abstract approach to coherence, LNM 281), functors had been replaced by "shapes"---formal combinations of given functors---with comments to indicate that this replacement is necessary to give correct statements. It is not too hard to build examples that show that the usual coherence statements are false if stated for functors and not shapes. However, I have never seen an example in print. Does anyone know of a reference for an example?

For the curious, here is an example. Let $A_n$ be $n$-tuples of integers regarded as sequences of length $n$, and let $A_m\otimes A_n$ be $A_{m+n}$ obtained by concatenating the sequences. This makes $\otimes$ strictly associative. Build an (idiotic) isomorphism $\alpha$ from $(X\otimes Y)\otimes Z$ to $X\otimes(Y\otimes Z)$ by negating the values contributed by $X$ and $Z$ but not $Y$. Note that the two functors connected by $\alpha$ are identical even though the shapes are different. Now check that the usual pentagonal diagram connecting all five shapes on four variables commutes. Thus our mongrel $\alpha$ is coherent on shapes, but not on the functors themselves. All five functors on four variables are the same, but are connected by 16 different isomorphisms, corresponding to all the elements in $\mathbf (Z_2)^4$, that can be built from $\alpha$.