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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 $A\times B$ in $\mathscr{C}$ and a fixed natural isomorphism $$\sigma \colon \mathrm{Hom}(-,A\times B)\to \mathrm{Hom}(-,A)\times \mathrm{Hom}(-,B).$$

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

For any pair of morphisms $f^{A}\colon X\to A$ and $f^{B}\colon X\to B$, there exists a unique morphism $f\colon X\to A\times B$ 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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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.

Blockquote

In a category $\mathscr{C}$, the product of $A$ and $B$ is the pair of object $A\times B$ in $\mathscr{C}$ and a fixed natural isomorphism $$\sigma \colon \mathrm{Hom}(-,A\times B)\to \mathrm{Hom}(-,A)\times \mathrm{Hom}(-,B).$$

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

Blockquote

For any pair of morphisms $f^{A}\colon X\to A$ and $f^{B}\colon X\to B$, there exists a unique morphism $f\colon X\to A\times B$ 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 $\mathscr{C}$.

1

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.

Blockquote In a category $\mathscr{C}$, the product of $A$ and $B$ is the pair of object $A\times B$ in $\mathscr{C}$ and a fixed natural isomorphism $$\sigma \colon \mathrm{Hom}(-,A\times B)\to \mathrm{Hom}(-,A)\times \mathrm{Hom}(-,B).$$

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

Blockquote For any pair of morphisms $f^{A}\colon X\to A$ and $f^{B}\colon X\to B$, there exists a unique morphism $f\colon X\to A\times B$ 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 $\mathfrak{Set}^{\mathscr{C}^{\mathrm{op}}}$ and the world of morphisms $\mathscr{C}$.