ct.category theory - "Philosophical" meaning of the Yoneda Lemma

Here is an example on representable functors. Yoneda's lemma gives down-to-earth, morpshim oriented interpretation of representable functors, and vice versa.

I will explain this with an example.

In a category $mathscr{C}$, the product of $A$ and $B$ is the pair of object $Atimes B$ in $mathscr{C}$ and a fixed natural isomorphism

$$sigma colon mathrm{Hom}(-,Atimes B)to mathrm{Hom}(-,A)times mathrm{Hom}(-,B).$$

This definition of products only uses terminology of functors. By applying Yoneda's lemma, we arrive at a morphism oriented definiton of products. Yoneda's lemma says that there is a bijection

$$Psi colon mathrm{Hom}left( mathrm{Hom}(-,Atimes B),mathrm{Hom}(-,A)times mathrm{Hom}(-,B)right) to mathrm{Hom}(Atimes B,A)times mathrm{Hom}(Atimes B,B).$$
In particular, we apply this to $sigma$ and denote
$$Psi(sigma)=sigma(Atimes B)(mathrm{id}_{Atimes B})=(pi^{A}colon Atimes Bto A,pi^{B}colon Atimes Bto B).$$
Next, by applying the inverse of $Psi$, we compute
$$sigma(X)=Psi^{-1}left( Psi(sigma)right)(X):mathrm{Hom}(X,Atimes B)to mathrm{Hom}(X,A)times mathrm{Hom}(X,B)$$
$$fcolon Xto Atimes Bmapsto (pi^{A}circ f,pi^{B}circ f).$$
Since $sigma$ is a natural isomorphism, $sigma(X)$ is a bijection. This bijectivity is the usual definition of product based on morphisms (universality):

For any pair of morphisms $f^{A}colon Xto A$ and $f^{B}colon Xto B$, there exists a unique morphism $fcolon Xto Atimes B$ with $pi^{A}circ f=f^{A}$ and $pi^{B}circ f=f^{B}$.

I think the Philosophy behind Yoneda's lemma is that, it connects the world of functors (and natural transformations) $mathfrak{Set}^{mathscr{C}^{mathrm{op}}}$ and the world of morphisms $mathscr{C}$.

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