Empty Product is Terminal Object
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Theorem
Let $\mathbf C$ be a metacategory.
Suppose $\mathbf C$ admits a product $\prod \O$ for the empty set.
Then $\prod \O$ is a terminal object of $\mathbf C$.
Proof
By definition, $\prod \O$ is the limit of the empty subcategory $\mathbf 0$ of $\mathbf C$.
The result follows from Terminal Object as Limit.
$\blacksquare$