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 $(x,a)$, $a\in C$, $x\in F(a)$. 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 $a\in C$ 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 $C$ $$\int \mu: El(p;\cdot) \simeq p/C \to El(F)$$
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 $1:p\to p$. This reminds that an automorphism of Galois covering is uniquely defined by choosing the image of one element in the fibre, thus $$Aut(M\stackrel{p}{\to} N) = p^{-1}(x),\;x\in N$$
A morphism of Galois coverings $f:X\to Y$ 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.