= Canonical colimit presentation of a presheaf
{title2=$P\cong\operatorname{colim}_{(c,x)\in\int P}H_c$}
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.
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