ct.category theory - What is Yoneda's Lemma a generalization of?

I like to view Yoneda's lemma as a generalization of the description of Galois coverings in topology. To any functor $F: C to Set$ we can associate its category of elements $El(F)$. Its objects are pairs $(x,a)$, $ain C$, $xin F(a)$. A morphism $f:(x,a)to (y,b)$ is a morphism $f_*: a to b$, such that $F(f_*)(x) = y$. Such a category is equipped with a natural projection $Q_F : El(F) to C$, sending $(x,a)$ to $ain C$ and a morphism in $El(F)$ to the underlying morphism in $C$. Then it is easy to see that a natural transformation $mu: (p;cdot) to F(cdot)$ is just the same as a morphism of fibrations over $C$ $$int mu: El(p;cdot) simeq p/C to El(F)$$

This is an example of Grothendieck's construction, applied to set-valued functors. It is itself a categorical version of the correspondence between sheaves of sets and their etale spaces in algebraic geometry.

Consider for example $Nat[(p;cdot);(p;cdot)]$. By Yoneda's lemma it equals to $Hom_C(p;p)$. This is exactly the fibre of $p/C$ over $C$ under the Grothendieck's construction for $(p;cdot)$. The whole automorphism of $(p;cdot)$ is thus determined by the image of $1:pto p$. This reminds that an automorphism of Galois covering is uniquely defined by choosing the image of one element in the fibre, thus $$Aut(Mstackrel{p}{to} N) = p^{-1}(x),;xin N$$

A morphism of Galois coverings $f:Xto Y$ with $X$ connected is likewise uniquely determined (if it exists) by choosing some element of a fibre of $Y$. If $X$ is contractible, then a morphism always exists. This means that slice categories $p/C to C$ are actually similar to contractible fibrations. I don't know how far the analogy goes, but via the classifying space functor slice categories really map to contractible spaces, because they have initial objects.

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