Canonical colimit presentation of a presheaf ($P\cong {\mathrm{colim}}_{(c,x)\in \int P}{H}_c$)

For a categorical presheaf $P$ on a small category, its category of elements has objects $(c,x)$ with $x\in P(c)$ and arrows $u:(c,x)\to (d,y)$ satisfying $P(u)y=x$. The canonical natural transformation $H_c\to P$ sends $v:a\to c$ to $P(v)x$. These maps exhibit $P$ as the displayed colimit. At $a$, a representative $(c,x,v)$ maps to $P(v)x$, with inverse $z\mapsto (a,z,1_a)$. The arrow $v:(a,P(v)x)\to (c,x)$ identifies every representative with its canonical inverse image, proving the bijection and the colimit universal property.