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A monoidal category has a natural isomorphism $\alpha_{A,B,C}\colon(A\otimesB)\otimes C\to A\times (B\otimes C)$ called the associator. In the case wherethe objects $(A\otimes B)\otimes C$ and $A\otimes(B\otimes C)$ are equal, it isnatural to ask whether this map may be taken to be the identity. Isbell gave anargument for an obstruction to strictifying the component of the associator ofa cartesian monoidal category at an object $C=C\times C$. This argument hasbeen widely reproduced and is commonly misunderstood as demonstrating thatnaturality of the associator is the true obstruction to strictification. Weconsider the hidden hypothesis in this argument, give a new argument notdependent on naturality but on the hidden hypothesis, and finally show thatnaturality alone is not the issue -- rather the crux of Isbell's argumentinvolves a hidden assumption concerning the product cones. Through thisanalysis we also resolve that there can be no general obstruction tostrictifying a component of the of the associator, even at such an object $C$,in a cartesian monoidal category.

A common misinterpretation of Isbell's obstruction to monoidal strictification