Let $\mathcal C$ be a category with products. Find a reasonable candidate for the universal property that the product $A\times B\times C$ of three objects of $\mathcal C$ ought to satisfy, and prove that both $(A\times B)\times C$ and $A\times(B\times C)$ satisfy this property.
I am trying to parse this statement, but ultimately becoming confused.
Isn't $A\times B\times C$ anyway the universal object in the category $\mathcal C_{A,B,C}$? Is there some other reasonable candidate that one should think of?
I will try and explain what I understand from $(A\times B)\times C$. First we determine the universal object in the category $\mathcal C_{A,B}$. Then we determine the universal object in the category of objects mapping to both $A\times B$ (which is the universal object of $\mathcal C_{A,B}$) and $C$. All the diagrams should commute, etc.
Is my understanding accurate? How do I show that $(A\times B)\times C$ satisfies this universal property?
Assuming that $C_{A,B,C}$ means what I think it does, item 1 of your question is the correct answer for "find a reasonable candidate $\dots$", and I can't think of another reasonable candidate (though I might just not be imaginative enough). You understanding of $(A\times B)\times C$ is correct also. To show that it satisfies the definition of $A\times B\times C$, let $p_{A\times B}$ and $p_C$ be its projections to $A\times B$ and to $C$; also, let $q_A$ and $q_B$ be the projections of $A\times B$ to $A$ and to $B$. Then $q_A\circ p_{A\times B}$, $q_B\circ p_{A\times B}$, and $p_C$ are the three projections of $(A\times B)\times C$ that should have the universal property that defines $A\times B\times C$. Verifying the universal property is routine, though a bit tedious, so I won't write out the details here.