I'm familiar with monoidal categories and want to make sure I completely understand the definition. One thing I keep getting stuck on is the natural isomorphism between $(X\otimes Y)\otimes Z$ and $X\otimes (Y \otimes Z)$, and in what sense it's a natural transformation. This seems to be quite a subtle and important point, but it seems to be almost universally glossed over, and I'm wondering if I'm missing some background that would make it obvious.
Anyway, it's clear that $(-\otimes -)\otimes -$ and $-\otimes (- \otimes -)$ are both functors into the category in question, $\mathscr{C}$, and that we are looking for a natural transformation between them. The issue is that, it seems to me, $(-\otimes -)\otimes -$ is a functor from $(\mathscr{C}\times \mathscr{C})\times \mathscr{C}$, whereas $-\otimes (- \otimes -)$ is a functor from $\mathscr{C} \times (\mathscr{C} \times \mathscr{C})$.
From the definition of the product of categories these seem to be different, since one's objects are pairs $((X,Y),Z)$ and they other's objects are pairs $(X,(Y,Z))$. It's clear that they're isomorphic, but it seems like we should have to do the exact same thing we do in the definition of a symmetric monoidal category, namely to define an explicit isomorphism (in Cat) between $(\mathscr{C}\times \mathscr{C})\times \mathscr{C}$ and $\mathscr{C} \times (\mathscr{C} \times \mathscr{C})$, call it $A_\mathscr{C}$.
Then we have functors $$((-\otimes -)\otimes -):(\mathscr{C}\times \mathscr{C})\times \mathscr{C}\to \mathscr{C}$$ and $$(-\otimes (-\otimes -)):\mathscr{C}\times (\mathscr{C} \times \mathscr{C})\to \mathscr{C},$$ and the natural transformation we're looking for is not between $(-\otimes -)\otimes -$ and $-\otimes (-\otimes -)$ but between $((-\otimes -)\otimes -)\circ A_\mathscr{C}$ and $-\otimes (-\otimes -)$, both of which go from $\mathscr{C} \times (\mathscr{C} \times \mathscr{C})$ to $\mathscr{C}$.
After that explanation, my questions are
Am I on the right track here, and all this machinery exists but is hidden in a seemingly innocent phrase like "natural in $X$, $Y$ and $Z$"?
Or is there some other, different machinery that lets us treat $(\mathscr{C}\times \mathscr{C})\times \mathscr{C}$ and $\mathscr{C} \times (\mathscr{C} \times \mathscr{C})$ as not just isomorphic but equal, and it's that machinery that's hiding behind the definition?
Or am I barking up the wrong tree completely and making things more complicated than they need to be? If so, how is the issue I describe resolved?
I'm sorry if the question seems a bit nitpicky, but the definitions of these things are nitpicky by their nature, and when self-learning it's not always obvious which of these little details one needs to pay attention to.
Edited to add: Andreas Blass' answer below is great, but in trying to teach this to some colleagues, I'm finding it very difficult to use that approach. Of course we're fully justified in treating the categories $(\mathscr{C}\times\mathscr{C})\times\mathscr{C}$ and $\mathscr{C}\times(\mathscr{C}\times\mathscr{C})$ as the same - that's completely obvious to me now - but the thing that justifies that is the fact that $\mathrm{Cat}$ in this context is a monoidal category.
So it seems that if I don't want to wave my hands about the domains of $({-}\otimes{=})\otimes{\equiv}$ and ${-}\otimes({=}\otimes{\equiv})$ then I first have to first explain Mac Lane's coherence theorem for the special case of $\mathrm{Cat}$, and then use that result to define the associators for a general monoidal category in general, and then state it again for the general case. I just can't escape the feeling that there has to be a better way to explain this.
Unfortunately, this question has too many answers --- depending on whether the answer comes from a set theorist or a category theorist, and also depending on whether you ask the question and get an answer explicitly or whether you just observe what people do in practice. Here's how the situation looks to me:
Set theorist: Since you've asked explicitly and thus made me pay attention to the issue, the answer is, strictly speaking, no; $(\mathcal C\times\mathcal C)\times\mathcal C$ and $\mathcal C\times(\mathcal C\times\mathcal C)$ are not literally the same but only canonically isomorphic. But in practice, I suppress mention of this canonical isomorphism, so in effect I act as though they are equal.
Category theorist: Products are defined only up to isomorphism, so it doesn't make sense to ask whether $(\mathcal C\times\mathcal C)\times\mathcal C$ and $\mathcal C\times(\mathcal C\times\mathcal C)$ are equal. We can only ask whether they are isomorphic, and the answer to that is yes. In fact, there is a unique isomorphism that commutes with the obvious projections (the ones sending the $k$-th $\mathcal C$ in the notation $(\mathcal C\times\mathcal C)\times\mathcal C$ to the $k$-th in the notation $\mathcal C\times(\mathcal C\times\mathcal C)$ for each $k=1,2,3$). These associativity isomorphisms are special cases of more general $(\mathcal C\times\mathcal C')\times\mathcal C''\cong\mathcal C\times(\mathcal C'\times\mathcal C'')$, which are natural in all three variables. Furthermore, the (more general) associativity isomorphisms obey the well-known coherence conditions for a monoidal structure, and so the coherence theorems usually allow us to suppress mention of these isomorphisms. In effect, we can act as though $(\mathcal C\times\mathcal C)\times\mathcal C$ and $\mathcal C\times(\mathcal C\times\mathcal C)$ are the same.
Notice that both the set theorist and the category theorist end up, in practice, with what you've observed; $(\mathcal C\times\mathcal C)\times\mathcal C$ and $\mathcal C\times(\mathcal C\times\mathcal C)$ are treated as if they were the same. The category theorist invokes actual theorems (Mac Lane's coherence theorem) to justify this practice; the set theorist is more inclined to ignore the issue or treat it as obvious. I have not thought carefully about whether coherence theorems fully justify treating $(\mathcal C\times\mathcal C)\times\mathcal C$ and $\mathcal C\times(\mathcal C\times\mathcal C)$ as the same or whether there's some remnant of ignored-or-obvious even in the category theorist's view.