The same question is given here Categorical products are associative, however, one of the answers uses concepts not introduced yet in my book, and the other provides a link (https://compose.ioc.ee/categoryTheory2020/week3/week3.pdf, "Diagram Chasing") which to me is still unclear. I am trying to prove $A \times (B \times C) \cong (A \times B) \times C$
Basically, the answer in the link states that we can form the following four commuting diagrams:
From these we have the diagram:
Then the author concludes that because this arrow is unique, it must be the identity arrow. This is the part that I do not follow. Where in the proof have we concluded that $q_2 \circ q_1$ makes the last diagram commute? Once that is established then I understand how we can conclude $q_2 \circ q_1 = 1_{A(BC)}$.