Theorem. [dt-002J]

There is an isomorphism between pairs of continuous functions and continuous functions that output pairs: (A \twoheadrightarrow B) \times (A \twoheadrightarrow C)\; \simeq \; (A \twoheadrightarrow (B \times C))

Proof.

Proof follows from the universal property after setting up the isomorphism with functions f(g_0,g_1) = \langle g_0 , g_1 \rangle and f^{-1}(h) = (\pi _0 \circ h, \pi _1 \circ h).