Can any object in a Cartesian Closed Category regarded as a binary product?

Question

Suppose there is an object $Z$ in a CCC, can we regard it as $Z \times 1$ where $1$ is the terminal product? Why it is true or otherwise?

Answer 1

Yes, up to isomorphism. You don't even need your category to be cartesian closed, all you need is for it to have a terminal object.

To see this, it is easy to prove that $Z$ satisfies the universal property of the product $Z \times 1$ using the fact that there is a unique morphism from any object to $1$. Since $Z$ satisfies the universal property, it must be isomorphic to $Z \times 1$.