I am trying to understand the definition of Ends and CoEnds.
I was looking at some examples where I found the following case because of which I have become more confused.
So, here is the example:
Let $G \in [\textbf{C}^{op} \to \textbf{Set}]$, i.e. Let G be a presheaf. Now, the example talks about: $\int_{B}GB$.
This does not make sense to me. From what I know, Ends are defined on functors from $\textbf{C}^{op} \times \textbf{C}$ to another category D which in this case will be Set. How can one even write $\int_{B}GB$?!
I have tried thinking about this for a while and I thought maybe we can apply Yoneda's Lemma on $GB$. So, we will be able to write: $GB \simeq \hat{\textbf{C}}(\textbf{C}(-, B), G)$. So, we will be able to write: $\int_{B}GB \simeq \int_{B}\hat{\textbf{C}}(\textbf{C}(-, B), G)$.
Now, is $\hat{\textbf{C}}(\textbf{C}(-, B), G)$ a functor from $\textbf{C}^{op} \times \textbf{C}$ to Set?
I think I am missing something very simple or basic here. Any help with this would be great.
Thank you.
$\int_BGB$ stands for $\int_B\tilde G(B,B)$ where $\tilde G:{\bf C}^{op}\times {\bf C}\to{\bf Set}$ is defined as $$\tilde G(A,B):=GA\ \text{ and }\ \tilde G(f,g):=Gf\,.$$ One readily sees that it's indeed a functor, e.g. by observing that $\tilde G=G\circ\pi_1$ where $\pi_1:{\bf C}^{op}\times {\bf C}\to{\bf C}^{op}$ is the projection on the first coordinate.
As commented by Fosco, one can prove that $\int_B\tilde G(B,B)$ is just the limit of $G$ in $\bf Set$.