This was already mentioned, but the Yoneda lemma (and its dual, as well as extensions to enriched and higher categories) is perhaps one of the most important theorems of category theory. There are proofs of serious theorems which can be boiled down to repeated different uses of Yoneda [1]. It could probably be seen as the categorical analogue of Cauchy-Schwartz.

Lawvere refers to the lemma as the Cayley-Dedekind-Grothendieck-Yoneda Lemma, giving an idea of the scope of its uses and users. The question What is Yoneda's Lemma a generalization of? and its answers give an idea of results that follow from Yoneda.

[1] Urs Schreiber could no doubt recall some, I remember him emphasising this fact once.

This was already mentioned, but the Yoneda lemma (and its dual, as well as extensions to enriched and higher categories) is perhaps one of the most important theorems of category theory. There are proofs of serious theorems which can be boiled down to repeated different uses of Yoneda [1]. It could probably be seen as the categorical analogue of Cauchy-Schwartz.

Lawvere refers to the lemma as the Cayley-Dedekind-Grothendieck-Yoneda Lemma, giving an idea of the scope of its uses and users. The question What is Yoneda's Lemma a generalization of? and its answers give an idea of results that follow from Yoneda.

[1] Urs Schreiber could no doubt recall some, I remember him emphasising this fact once.