I was reading Fosco's notes coend calculus, so far I am at Ninja-Yoneda lemma.
The density formula for pre-sheafs says that,
$${K} V\cong \int^{U\in C} K U\times \textstyle\text{Hom}_{C}(V,U)\cong\displaystyle\int_{U\in C}(K U)^{\textstyle\text{Hom}_{C}(U,V)}.$$ What's the intuitive meaning behind this? I view ends and coends as some sort of averaging over all morphisms in the category, as suggested by Fosco.
The reason for asking this is I need to get a feeling for what is happening, otherwise it remains mysterious as to how someone came up with some random formula like this (which you can prove yes, but then I will have no clue how they got to it)
The proof seems to be to use a chain of isomorphism that lets us sneak in Yoneda lemma.
The last step seems to be some application Yoneda lemma, but I can't figure it out. I have the following, but not sure how this is $Nat(C(x,-),Sets(K-,y))$. Best I could do is,
$$\textstyle\text{Hom}_{\mathbf{Sets}}(K V, W)\cong \textstyle\text{Hom}_{\mathbf{Sets}}\big(\textstyle\text{Hom}_{\mathbf{Sets}^{{C}^{op}}}(h^V,K ),W\big) $$
Edit: Skip the last part, I was applying Yoneda to the functor $K$ (Rookie mistake I guess), I should have applied it to $F(-)=\text{Hom}_{\mathbf{Sets}}(K(-),W)$, which will give me the required, $$\text{Hom}_{\mathbf{Sets}}(K(x),W)\cong Nat(h^x,F(-))$$