Canonical Isomorphism before Category Theory

Question

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?

Answer 1

Often, mathematical constructions and theorems include a lot more details than are actually stated. For example, consider the first isomorphism theorem of groups, which is usually stated:

If $\varphi : G \to H$ is a group homomorphism, then $G / \ker \varphi \cong \operatorname{im} \varphi $

but the first isomorphism actually says something stronger:

If $\varphi : G \to H$ is a group homomorphism, then the mapping that sends the congruence class of $g$ to $\varphi(g)$ is as well-defined isomorphism of groups

As another example, if $N$ is a normal subgroup of $G$, we often describe taking the quotient as producing a group we call $G/N$. But ths construction produces more than that: it also produces a group homomorphism $G \to G/N$ sending $g$ to its congruence class.

Commonly, the phrase "canonical foo" is used in a situation where filling in these unspoken details leads directly to the construction of a foo, and it's that specific foo that is meant.

(disclaimer: I've not read the source you're referring to)