Check that the product \(X \xleftarrow{p_X} X \times Y \xrightarrow{p_Y} Y\) is terminal object in \(\mathbf{Cone}(X,Y)\)
The existence and uniqueness of the morphism to the product object in the product definition are the conditions of being a terminal object in \(\mathbf{Cone}(X,Y)\). The maps of the terminal object to \(X\) and \(Y\) are the projection maps of the product.