Every Presheaf Is Colimit of Representables

Question

I am working on the proof that each presheaf $F\in \mathbf{Set}^{\mathcal C^{op}}$ is the colimit of $\mathbf{y}\circ \pi\colon \int_{\mathcal{C}}F\to \mathbf{Set}^{\mathcal C^{op}}$ where $\int_{\mathcal{C}}F$ is the category of elements of $F$, $\pi\colon (x,C)\mapsto C$ is the natural projection and $\mathbf{y}$ is the Yoneda embedding. In the course of the proof, I showed there was an embedding of $\int_{\mathcal C}F$ into the slice category $\mathbf{Set}^{\mathcal C^{op}}/F$ to obtain a cocone with vertex $F$. Then using the Yoneda lemma, for any other cocone $(p_x\colon \mathbf{y}C\to P)_{(x,C)}$ under $\mathbf{y}\circ \pi$, I constructed the natural transformation $\alpha$ componentwise, using the Yoneda lemma. However, I cannot for the life of me see where I needed the arrows of $\int_{\mathcal{C}}F$ in the proof. I see how the cocone is compatible with the arrows, but can't see how they are not extraneous to the proof.

So my question boils down to: Can someone provide a counterexample of a presheaf $F$ that is not a colimit of the functor $\mathbf{y}\circ\pi$ where the domain is the discretization of the category of elements of $F$, i.e., forgetting the nontrivial arrows?

Answer 1

There is a conceptual proof which goes along these lines:

1. The Yoneda Lemma entails that for any presheaf $F\colon \mathcal C^\text{op}\to \bf Set$, there is a canonical isomorphism $F\cong \int^X\hom(-,X)\times FX$ (see here);
2. You can interpret the former result as "any presheaf is canonically a weighted colimit";
3. $\bf Set$-weighted colimits correspond to colimits over the category of elements of the weight: $$ \text{lim}^{W} F\cong \text{lim}_{(c,x)\in \text{el}(W)}(F \pi) $$ The proof goes by nonsense (allow me to be sloppy but evocative, it's up to you to fill the details): $$ \begin{align*} \int^{c\in \bf C}Fc\times {Wc} &\cong {\rm coeq}\Big(\coprod_{c\to c'}Fc'\times {Wc} \rightrightarrows \coprod_{c\in \bf C}Fc\times {Wc}\Big)\\ &\cong {\rm coeq}\Big( \coprod_{c\to c'}\coprod_{x\in Wc}Fc'\rightrightarrows \coprod_{c\in\bf C}\coprod_{x\in Wc} Fc \Big)\\ &\cong {\rm coeq}\Big( \coprod_{c\to c'\in \text{el}(W)} Fc'\rightrightarrows \coprod_{(c,x)\in \text{el}(W)} Fc \Big)\\ &\cong \varinjlim_{(c,x)\in\text{el}(W)}(F\circ\pi) \end{align*} $$
4. Points 1+3 reads now as $F\cong \text{lim}^F\textbf{y}$, so by point 3 you get the result.