What do the usual set operations (intersection, union, complement,…) and the structure of first-order formulas have in common? In this talk, we will see how both can be described in the context of category theory through the notion of “Doctrines”. After a brief review of first-order logic, we will compare two worlds: we will observe that the set of all subsets of a given set and the set of all first-order formulas that depend on some fixed free variables can be endowed with the structure of a Boolean algebra. This will allow us to show that the powerset functor and indexed families of first-order formulas are both examples of first-order Boolean doctrines. These are categorical structures where logical connectives and quantifiers are interpreted as algebraic operations and adjunctions. Finally, I will give a hint about my current research topic, where I am exploring the concept of “quantifier-free formula” in the doctrinal setting.