$\mathcal{E}(-,E)○A$ has a left adjoint

Theorem: If $$A:\mathbf{C} \rightarrow \mathcal{E}$$ is a functor from a small category $$\mathbf{C}$$ to a cocomplete category $$\mathcal{E}$$, the functor $$R$$ from $$\mathcal{E}$$ to presheaves given by, $$R(E):C \rightarrow Hom_{\mathcal{E}}(A(C),E)$$ has a left adjoint $$L:\hat{\mathbf{C}} \rightarrow \mathcal{E}$$ defined for each presheaf $$P$$ in $$\hat{\mathbf{C}}$$ as the colimit $$L(P)=colim(\int P \xrightarrow{\pi_{p}} \hat{\mathbf{C}} \xrightarrow{A} \mathcal{E}).$$

I'm trying to understand this theorem. What is the intuitive idea behind it?

A few things: this left adjoint is, like any left adjoint, cocontinuous. Less obviously, $$L$$ restricts to $$A$$ itself, upon composing with the Yoneda embedding. This requires showing that any object is the colimit of its own representable functor's category of elements, a good thing to prove for yourself if you don't already know it. Finally, $$L$$ is the only possible cocontinuous functor extending $$A$$ in this way, which follows from the fact that every presheaf is the colimit of representables indexed by its category of elements. This is a harder but extremely important result (generalized by the theorem you've stated) sometimes called the co-Yoneda lemma.
In short, this result extends to show that cocontinuous functors out of $$\widehat{\mathbf C}$$ are "the same thing" as functors out of $$\mathbf C$$, precisely, the categories thereof are equivalent. This means the presheaf category is the free cocompletion of the category, in a way closely analogous to any free construction in classical abstract algebra--except that the presheaf category is large in size compared to the original category.
• This result is the so-called co-Yoneda lemma, if you take $A$ to be the Yoneda embedding. Jun 2, 2021 at 22:33