Covariant density presentation ($F\cong {\mathrm{colim}}_{(c,x)\in (\int F{)}^{\mathrm{op}}}\mathcal{C}(c,-)$)

For a set-valued covariant functor on a small category, the Yoneda lemma sends the element $x\in F(c)$ to the transformation $\mathcal{C}(c,-)\to F$. These transformations form a cocone indexed by the opposite category of elements. At $a$, send the representative $(c,x,u:c\to a)$ to $F(u)x$. It is surjective using $(a,z,1_a)$; two representatives of $z$ are identified through that same object of the category of elements. This proves the displayed presentation pointwise.