i think the horizontal arrows should've been \mapsto and not \to, since the vertical diagrams are not in the same category
Riley: What do you mean by "the logic of all type theories is constructive?"
In the type theories that you are working with, right? Because I thought that was a general claim about type theories
so a sum type is a coproduct ?
(Just for the record, I had in mind simple as well as ramified type systems -- typically their logic is classical. But you're point is right anyway.)
can we define an internal logic to a category that doesn't have all finite limits, since in that case, we dont necessarily have a subobject classifer ?