In Set, the categorical product of two sets is our usual cartesian product.
The projections are \(x \xrightarrow{p_x}(x,y)\xrightarrow{p_y}y\)
The unique morphism from some \(X \xleftarrow{f} C \xrightarrow{g} Y\), the unique map \(C \xrightarrow{!}X \times Y\) is given by \((f\times g)(c):=(f(c),g(c))\).