TITLE: What use is the Yoneda lemma?
QUESTION [61 upvotes]: Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse concepts. Many of the tools of category theory seem quite useful to me, such as Mitchell's embedding theorem, which allows one to prove theorems in any abelian category using diagram chasing. It lets me the ability to treat lots of objects I would not otherwise feel comfortable with as if they were modules over some ring; in essence, I feel like I've gained some tools from it.
However, I simply cannot see where to apply the Yoneda lemma to some useful end. This is not to say that I don't think it is a very pretty lemma, which I do, or that I do not appreciate the aesthetic of being able to study an object in a category by looking at the morphisms from that object, which I also do. And I do find it useful to consider the modules over a ring rather than the ring itself when studying that ring, or to treat groups as subgroups of permutation groups, which are the two applications I've heard of the Yoneda lemma. The problem is that I already knew these things could be done. Essentially, I don't feel like I've gained any tools from the Yoneda lemma. 
My question is this: how can the Yoneda lemma be applied to make problems more approachable, other than in cases like those I have listed above which can easily be treated without a general result like the Yoneda lemma? Basically, what new tools does it give us?

REPLY [12 votes]: This answer is much more naive than the others, but I'll post it anway.
The most standard application of Yoneda's Lemma seems to be the uniqueness, up to unique isomorphism, of the limit of an inductive or projective system. 
See for instance this pdf file by Pierre Schapira, or, more classically, the very beginning of:
Grothendieck, Alexander, Éléments de Géométrie Algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie. Publications Mathématiques de l'IHÉS, 11 (1961), p. 5-167. 
I don't see how to prove this uniqueness without using (explicitly or implicitly) Yoneda's Lemma.
EDIT A. Let me try to explain more precisely what I have in mind by taking the example of final objects. (This example is given by Grothendieck in Section $(8.1.10)$ page $8$ of the linked text.)
First, let $S$ be the category of sets, and let $C$ be any category. Denote by $C(x,y)$ the set of $C$-morphisms from $x$ to $y$. Let $F:C\to S$ be a contravariant functor. 
Say that a representation of $F$ consists of an object $a$ of $C$ together with an isomomorphism $F\simeq C(?,a)$. (By an enormous abuse of language, one usually expresses this by saying that $a$ represents $F$.)
Yoneda's Lemma tells us that, to each pair $F\simeq C(?,a)$ and $F\simeq C(?,b)$ of representations of $F$ is attached a perfectly defined isomorphism $a\simeq b$, and that the isomorphisms attached to any three representations of $F$ form a commuting triangle. 
So, we won't get in trouble if, among all representations of $F$, we pick one of them, call the object involved the representative of $F$, and denote it by a symbol like $a_F$. 
This is something we all do almost always.
(This is explained by Grothendieck in Section $(8.1.8)$ page $7$.)
Going back to our final object, let $s$ be a set with a single element, and $F:C\to S$ the contravariant functor attaching $s$ to any object of $C$. If $F\simeq C(?,\omega)$ is an isomorphism, we say, that $\omega$ is the final object of $C$. 
This of course applies (as pointed out by Grothendieck in Section $(8.1.9)$ page $7$) to any projective system. 
EDIT B. This is just a complement to t.b.'s interesting comment, which mentions this answer of ineff's about adjoint functors. 
The interested reader might take a look at Section 1.5 pages 38 to 41 about adjoint functors in this pdf scan taken from the Springer edition of EGA I. More precisely, the scan contains pages 19 to 41 of 
Éléments de Géométrie Algébrique I, Volume 166 of Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, A. Grothendieck, Jean Alexandre Dieudonné, Springer-Verlag, 1971.
Apart from this Section on adjoint functors, these pages are essentially contained into the part of the IHES edition of EGA referred to in the previous edit. 
This scan (I think) can be read with almost no prerequisite, and I don't know any other part of Grothendieck's work for which this can be said.