Proof that the categorical product is associative

Question

Despite the title, I think I want to show a bit more than associativity (up to isomorphism). This is Exercise 1.35(a) in Strom's Modern Classical Homotopy Theory (https://books.google.co.uk/books?id=Q4GDAwAAQBAJ)

Show that if one of the prducts $X\times(Y\times Z)$ and $(X\times Y)\times Z$ exists in $\mathcal{C}$, then so does the other, and they are isomorphic. (No special assumptions on $\mathcal{C}$.)

I don't understand how the universal properties of, say, the products of $X$ and $Y\times Z$, and of $Y$ and $Z$, imply that $X\times Y$ exists. I don't know what a "candidate" object for this product would be.

Answer 1

This is not true in this generality. We have to assume existence of both $X\times Y$ and $Y\times Z$ for this exact statement.

A counterexample: take the partially ordered set on $\{0,\,a,b,c,x,y,z\}$ by defining relations $$a\le x,\ b\le x,\ a\le y,\ b\le y,\ c\le y,\ c\le z\,,$$ and $0\le$ all.
In a poset, direct product is greatest lower bound, hence there is no $x\times y$, but $y\times z=c\ $ and $x\times c=0$, so $x\times(y\times z)$ exists.

However, the following similar statement holds about ternary direct product, i.e. the limit of the diagram shaped of $3$ objects:

If either $(X\times Y)\times Z$ or $X\times(Y\times Z)$ exists in a category $\mathcal C$, then the ternary product $X\times Y\times Z$ exists and is isomorphic to it.