Exercise. [dt-0028]

Prove that the three properties in § [dt-0027] are equivalent to the universal property for products

(\pi _0 \circ h = f \land \pi _1 \circ h = g)\ \text {iff}\ h = \langle f , g \rangle

Solution.

Showing in both directions:

The three properties follow from the universal property: The first two properties follow from the universal property where $h$ is set to $\langle f , g \rangle$ . The third property follows from the universal property where $f$ is set to $\pi _0\circ h$ and $g$ is set to $\pi _1\circ h$ .
The universal property follows from the universal property: In the $\Rightarrow$ direction of the universal property follows from the third property, and the $\Leftarrow$ direction follows from the first two.