Recovering the structure of an object from its morphism:Yoneda Lemma

Question

I've heard that Yoneda lemma informally states that one can recover the internal structure of an object by looking at the morphism coming out from that object. But this is not clear to me from the statement of the lemma. How can somebody recover the internal structure of an object by looking at the morphism coming out from that that object using Yoneda lemma?

Answer 1

Here's a cute, intuitive interpretation of the Yoneda lemma: suppose I have a topological space $X$, but I am not telling you what $X$ is. You have to figure it out. However, you are allowed to choose any space $Y$, and "probe" the unknown space using the space $Y$. In other words, I give you access to the functor $\hom(-, X)$. Then, using your probes, you can recover $X$ up to isomorphism.

(I'm assuming you're not just looking for a proof of the Yoneda lemma, which you can find anywhere.)