A grammar $𝐴$ is unambiguous if for all grammars $𝐵$ and maps $𝑒,𝑒\prime :𝐴⊢𝐵$ we have $𝑒=𝑒\prime$.

In a category with terminal objects, this is equivalent to unambiguity defined via subterminality.

Definition 1. Unambiguity as Subterminality unambiguity-as-subterminality

A grammar $𝐴$ is unambiguous if the unique map into the terminal object is a monomorphism. That is, $𝐴$ is a subobject of $\top$.