Prove that the product $X \times Y$ is exactly the same as the terminal object in $\bf{Cone}(X, Y)$

Question

I'm not sure if my thought is complete yet, but is the indexing category necessarily the discrete category $\bf{2}$?

Answer 1

Hint: A morphism $Z \to X \times Y$ is uniquely specified by a pair of morphisms $Z \to X$ and $Z \to Y$, and vice versa. What, by definition, are the morphisms into the terminal object of the category of cones over $X$ and $Y$ (with no diagrammatic arrows between them)?