**JB:** Okay, let’s talk more about how to do first-order classical logic using some category theory. We’ve already got the scaffolding set up: we’re looking at functors

You can think of as a set of predicates whose free variables are chosen from the set *S*. The fact that *B* is a functor captures our ability to substitute variables, or in other words rename them.

But now we want to get existential and universal quantifiers into the game. And we do this using a great idea of Lawvere: *quantifiers are adjoints to substitution*.