Empty Product is Terminal Object

Theorem
Let $\mathbf C$ be a metacategory.

Suppose $\mathbf C$ admits a product $\displaystyle \prod \varnothing$ for the empty set.

Then $\displaystyle \prod \varnothing$ is a terminal object of $\mathbf C$.

Proof
By definition, $\displaystyle \prod \varnothing$ is the limit of the empty subcategory $\mathbf 0$ of $\mathbf C$.

The result follows from Terminal Object as Limit.