I recently took a class in homological algebra and there I saw Yoneda's lemma. This says that if we have a locally small category $C$ and a functor $F:C\to\text{Set}$, then the natural transformations of $\text{Hom}(-,X)$ to $F$ are a set, more precisely we have a bijection between the natural transformations of $\text{Hom}(-,X)\to F$ and the set $FX$.
Now I am mostly interested in analytic aspects of mathematics but I feel that the above lemma is very widely applicable, to the point that I cannot grasp its potential. I was wondering, does anybody know of an application of Yoneda's lemma that produces an interesting result for some analytic objects, e.g. Banach spaces, Hilbert spaces?
I am new to this site so please tell me if my question is not suitable or if I must edit something.