# Monoidal category's associator as a natural isomorphism

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?

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.