Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 4 / e

By the canonical colimit presentation of a presheaf, writewith structure maps ${\alpha }_{c,x}:H_c\to P$. Since $P$ is a 0-presentable object, the canonical mapis bijective. Every element of a set-valued colimit has a representative in some component. In particular $1_P$ is the image of a transformation $s:P\to H_c$ for some $(c,x)$, soThus $P$ is a retract of $H_c$. The representable retract criterion from part (c), using the assumed equalisers in $\mathcal{C}$, now gives $P$ is representable. Only existence of the representative of $1_P$ is needed; no stronger uniqueness-of-factorization assertion about individual colimit injections has been assumed.