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

Use 0-presentable object in the specified sense: its hom-functor preserves all small colimits. For a representable $H_c$, the Yoneda lemma identifies this hom-functor with evaluation at $c$. Given any small diagram $P_j$, pointwise colimits giveThese are the canonical comparison bijections, including for an empty diagram. Therefore every representable presheaf is 0-presentable, or a small projective object in this sense.