I asked this question on M.SE, but didn't get any answers.

Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $X$ it's better to study *morphisms* into that object, $-\to X$. Apparently that's called the relative point of view.

**First question.** How is that principle applied in practice? What are some concrete examples in mathematics where the relative point of view is useful?

Wikipedia mentions the Riemann–Roch theorem and a similar MSE question mentions a theorem about coherent sheaves. Unfortunately, I don't know any algebraic geometry yet. Are there more down-to-earth applications of the relative point of view that an undergraduate can understand, say, in linear algebra, group theory, ring theory, Galois theory, or maybe even in basic category theory?

What are (some of) the most important theorems that feature the relative point of view?

I recently heard about the Yoneda lemma in category theory (I know the statement and can prove it). I know that it can be used to prove that two objects are isomorphic whenever they have the same universal property. In Awodey's category theory book, there's a concrete application of that: in categories with enough structure, $(A\times B)+(A\times C)\cong A\times (B+C)$. That proof is elegant, I agree. But it doesn't live up with the praise many people give to the Yoneda lemma, does it?

Maybe a more concrete application in non-category theory would help me to get convinced of the contrary. For instance, I read on Wikipedia (and elsewhere) that Grothendieck used the Yoneda lemma in his famous book EGA (which a lot of people seem to talk about). (In fact, it seems this was another insight of him: that Yoneda is useful.)

**Second question.** So what were Grothendieck's main applications of the Yoneda lemma in algebraic geometry? (In contrast to the first question, here it suffices for me to just know roughly what kind of statement he proved with the Yoneda lemma---rather than understanding it in detail, because I already know one application of the Yoneda lemma.)

**Third question.** Is the second question related to the first one, i.e., is there a connection between the relative point of view and the Yoneda lemma? (At least the Wikipedia page linked above mentions the Yoneda lemma.)

preciseexamples but this philosophy permeates throughout the main definitions/constructions in Grothendieck-style geometry : you never define things for schemes, but for maps of schemes $X\to Y$ . This allows to give some "inductive" kinds of proofs where to prove the absolute case you sometimes need to go through relative versions. For the third question, the answer is that the Yoneda lemma states that $X$ is completely recovered from $\hom(-,X)$, i.e. morphisms to $X$ $\endgroup$2more comments