 Universal property of Cartesian products

Given functions $f_i:Y\to X_i$, there is a unique function $g:Y\to X_1\times X_2$ whose coordinate projection maps are $f_1,f_2$. It is the displayed pointwise pairing. Equality of ordered pairs is equality of both coordinates, which proves uniqueness and makes the Cartesian product a categorical product in the Category of sets.