Is $X\times Y$ isomorphic to $Y\times X$? Let $\mathscr{C}$ be a category with finite products. Let $X,Y$ be objects of $\mathscr{C}$. Is it true in general that $X\times Y$ and $Y\times X$ are isomorphic objects?
 A: Yes. Given product diagrams
$$X \xleftarrow{p_1} X \times Y \xrightarrow{p_2} Y \qquad \text{and} \qquad Y \xleftarrow{q_1} Y \times X \xrightarrow{q_2} X$$
We obtain an isomorphism $\langle q_2, q_1 \rangle : Y \times X \to X \times Y$, with inverse $\langle p_2, p_1 \rangle$. Then, in fact,
$$X \xleftarrow{q_2} Y \times X \xrightarrow{q_1} Y$$
is a product diagram.

Remember, product diagrams — like anything defined by a universal property — are only defined up to isomorphism. $X \times Y$ is just a name we give to a particular choice of product of $X$ and $Y$ (usually with some implicit choice of projection morphisms), but there may be others. The above shows that $Y \times X$ is also a product diagram for the pair $X,Y$ (as well as the pair $Y,X$).
For example, in $\mathbf{Set}$, if $f : Z \to X \times Y$ is any bijection at all then
$$X \xleftarrow{\pi_1 \circ f} Z \xrightarrow{\pi_2 \circ f} Y$$
is a product diagram, where $X \times Y$ denotes the Cartesian product and $\pi_1, \pi_2$ are the usual projection maps. So for example any $6$-element set is a product of the sets $\{0,1\}$ and $\{0,1,2\}$ (but usually with different projection maps).
