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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?

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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.

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  • $\begingroup$ This result is the so-called co-Yoneda lemma, if you take $A$ to be the Yoneda embedding. $\endgroup$
    – Zhen Lin
    Jun 2, 2021 at 22:33
  • $\begingroup$ @ZhenLin Sure, good point. $\endgroup$ Jun 3, 2021 at 20:06

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