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A monoidal category is equipped with the associator $ \alpha_{A,B,C}:A\otimes (B\otimes C)\cong (A\otimes B)\otimes C$ which is a natural isomorphism.

What explicitly does it mean for this to be a natural isomorphism? A natural isomorphism is a natural transformation between functors and yet $A\otimes (B\otimes C)$ and $(A\otimes B)\otimes C$ are objects in a category. I read that the associator is natural in each argument. I interpret this as saying that, fixing objects $B$ and $C$, the functors $A\mapsto A\otimes (B\otimes C)$ and $A\mapsto (A\otimes B)\otimes C$ are naturally isomorphic. (With corresponding statements for the other two arguments). Is this the correct interpretation or am I missing something?

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This is almost correct. We can say that $(-)\otimes((-)\otimes(-))$ is a functor $M\times M\times M\to M$ and similarly $((-)\otimes(-))\otimes(-)$ is a functor $M\times M\times M\to M$; $\alpha$ is supposed to be a natural isomorphism of these functors. That's left implicit by "$A\otimes(B\otimes C)\cong(A\otimes B)\otimes C$".

Why is what you said almost correct? Well, we require $\alpha$ not just to be natural in $A$ (which is what you said) but in $B$ and $C$ as well. It's a straightforward exercise for multifunctors to see that naturality - as a whole functor over all of $M\times M\times M$ - is equivalent to naturality in each variable separately, with the others held fixed.

By the way, people are very, very quick to forget this but we require more than just a natural isomorphism: we require $\alpha$ to satisfy certain "coherence conditions". This is very important! See e.g. Mac Lane's chapter on monoidal categories to see what these are.

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