Suppose $\mathscr{C}$ is a category and $(A \times A,p,q)$ be the product of $A$ with itself, where $p$ and $q$ are the projection morphisms. My question is: can we always take $p=q$?
If so, is this true even when we take the fibre product of $A \rightarrow X$ with itself?
Is there some similar statement for (co-)limits in general?
It's important when learning category theory to remember that every concept is a generalization of something elementary, and nowhere is this more the case than for products. Projections out of products in abstract categories generalize coordinate functions from basic mathematics. Just as you must draw two different coordinate axes on the plane, you must use two different projections out of (almost) any product! A nice exercise would be to classify the choices of $A$ for which this works. They're extremely special.