I am studying Serre’s FAC and I am aware this work predates most formalization of category theory (or more correctly, before the language of category theory was ubiquitous, since the work was occurring at approximately the same time) although many common categorical notions were being described and used implicitly without the language of category theory.
In fact, something I have been doing as I work through FAC is summarizing each section in modern language. For example, Chapter 1, Section 1, Subsection 2 is essentially constructing and presenting the sheaffification functor, except for the fact that a sheaf to Serre is what is currently called an etale space, so his sheafification functor would take an arbitrary bundle over $X$ and return an etale space. Perhaps this should be called etalification. However, this functor is adjoint to the usual sheafification functor and both restrict to an equivalence of categories.
This whole task of summarizing subsections of FAC in modern language has me curious about something. In Chapter 1, Section 1, Subsection 2 in the statement of Proposition 3 Serre asserts that some objects are canonically isomorphic. In the modern language viewing the constructions of this subsection as functors it is actually true that there is a natural isomorphism between them. But without the language of functors, what precisely would Serre have taken canonically isomorphic to mean?