Yoneda Explained

Paolo Perrone's preprint Notes on Category Theory with Examples from Basic Mathematics brings a practical attitude toward introducing deep concepts to non-mathematicians. The author describes his plan: "Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept." His introduction concludes, "If you don't know what any of the above are, don't worry, everything will be introduced shortly. Let's now get started with the math."

And soon enough, in section 2.2 there's the Yoneda Lemma, proven and then summarized:

... if all ways of observing two objects X and Y coincide, then X and Y must be isomorphic. The Yoneda lemma says why: we can observe X in such a way that does not lose any information, namely, mapping it to itself via the identity, idX : XX. This is an observation that trivially sees the whole of X faithfully. Therefore, this one observation is sufficient to determine X uniquely. And conversely, every other observation, which possibly loses information, is obtainable from this one ...

and as Perrone points out earlier:

... each object of C is uniquely specified by the arrows into it (resp. out if it), up to isomorphism. The objects of a category can be uniquely defined in terms of the role they play in the category, in terms of their "interaction with the whole".

Caveat. This statement can be thought of as rather "philosophical", and it is similar to axioms in philosophy that one can assume true or not (such as the identity of indiscernibles). In category theory, however, this is a theorem, with a proof. It is true in every category.

(cf Yoneda Perspective (2018-10-03), If You Need a Theorem (2018-11-08), Reflect (20129-08-25), Yoneda Friend (2019-11-22), ...) - ^z - 2020-01-05