Clarifying statement of Yoneda's lemma

Question

I was understanding the statement of Yoneda's lemma from Hilton-Stambach's Homological algebra. I couldn't clarify few things in the statement before going into its proof.

Here $\mathfrak{C}$ is a category, and $\mathfrak{S}$ is the category of sets.

$\mathfrak{C}(A,-)$ is a covariant functor from $\mathfrak{C}$ to $\mathfrak{S}$. It takes an object $X$ in $\mathfrak{C}$ to the object $\rm{Mor}(A,X)$ in $\mathfrak{S}$, and a morphism $f:A\rightarrow Y$ to the morphism $f_*:\rm{Mor}(A,X)\rightarrow \rm{Mor}(A,Y)$, $f_*(\sigma)=f\sigma$.

Question 1. What is $\tau_A(1_A)$? Here $1_A$ is, perhaps, identity morphism on $A$.

Here $F$ is any functor from $\mathfrak{C}$ to $\mathfrak{S}$.

Question 2. What the author want to describe through this theorem? (After proof, I didn't find any comment on intuition for this theorem or its implication).

---

Please correct if there is symbolic typo mistake, since the questions are perhaps clear - just understanding terminologies in Proposition 4.1

Answer 1

Yoneda lemma is a very basic (and important) fact, and you should be able to find a lot of explanations and examples in the literature, math.stackexchange.com, mathoverflow.net, etc. So let me just comment on its use in the book of Hilton-Stambach (or any course of homological algebra).

The lemma is needed to deduce the Yoneda embedding as a particular case (which is Corollary II.4.2 in the book), and then the latter is used, for instance, to see certain properties of adjoint functors (e.g. the fact that adjoint functors determine each other up to isomorphism, which is Proposition 7.3 in the book; or the fact that right (resp. left) adjoints preserve limits (resp. colimits)). Many basic properties in homological algebra come from adjunctions between certain functors. You will see it if you read a bit more of the book.

And then, Yoneda makes a less expected appearance in §IV.10 ("Another Characterization of Derived Functors") of the same book! Hilton and Stambach present the following theorem:

Let $T$ be an additive right exact functor with values in the category of abelian groups. Then the natural transformations $\operatorname{Ext}^q (A, -) \Rightarrow T$ form an abelian group, which is naturally isomorphic to $L_q T (A)$.

$$\tag{*} [\operatorname{Ext}^q (A, -), T] \cong L_q T (A).$$

If you just started learning homological algebra, you probably don't know yet what $\operatorname{Ext}^q$ and $L_q$ are, but for $q = 0$ the formula above reads $$[\operatorname{Hom} (A,-), T] \cong T (A),$$ which is the usual Yoneda lemma (for additive functors)!

Things like (*) is actually what Yoneda had in his original paper (On the homology theory of modules, J. Fac. Sci. Univ. Tokyo. Sect. I. 7 (1954), 193–227), before Mac Lane popularized the statement that we know nowadays as the Yoneda lemma.

Answer 2

For Question 1: Yes, $1_A$ is the identity morphism of $A$. So it is an element of $\mathfrak C(A,A)$. The natural transformation $\tau$ consists of morphisms (subject to certain commutativity conditions) $\tau_X:\mathfrak C(A,X)\to F(X)$ for all objects $X$ of $\mathfrak C$. In particular, we have $\tau_A:\mathfrak C(A,A)\to F(A)$, so $1_A$ is in the domain of $\tau_A$ and thus $\tau_A(1_A)$ makes sense. It is an element of $F(A)$.

I won't try to answer Question 2, at least not now, because it's opinion-based ("what did the author want") and too broad (Yoneda's Lemma has lots of uses).