Definition of an $\mathcal{O}_X$-module generated by its global sections in Liu

Question

In Liu's Algebraic Geometry and Arithmetic Curves, the author says that an $\mathcal{O}_X$-module is generated by its global sections at $x \in X$ if

the canonical homomorphism $\mathcal{F}(X) \otimes_{\mathcal{O}_X(X)} \mathcal{O}_{X,x} \to \mathcal{F}_x$ is surjective.

I assume that what is meant by "the canonical homomorphism" is the map defined by $f \otimes [s] \mapsto [s] \star [f]$, where $\star$ is the action induced by the actions of $\mathcal{O}_X(U)$ on $\mathcal{F}(U)$ for all open $U$ containing $x$. This feels like the only reasonable definition, but I'm not sure I understand fully how it is "canonical" in the sense of being defined by a universal property?

I thought this should be the map induced by the universal property of tensor products, where we consider $\mathcal{F}(X)$, $\mathcal{O}_{X,x}$, and $\mathcal{F}_x$ as $\mathcal{O}_X(X)$-modules, but I don't see how there is a natural choice of $\mathcal{O}_X(X)$-module map from $\mathcal{O}_{X,x}$ to $\mathcal{F}_x$ if $\mathcal{F}_x$ is not a ring (so we can't choose to send $[s]$ to $[s] \star 1$).

So what is the right sense in which the homomorphism defined above is "canonical" (and do I even have the right definition?!)?

Answer 1

hunter in the comments:

Your definition is correct. "Canonical" can be a little overused/underdefined in math literature -- sometimes it means "morphism arising from an underlying natural transformation of functors" (which is I think what you are searching for) and sometimes it just means something informally "the only reasonable one that anyone would write down."