 Representable retract criterion

Suppose a small category has equalizers. If a categorical presheaf $P$ is a retract $P\stackrel{i}{\to }{H}_B\stackrel{p}{\to }P$, with $pi=1_P$, then $i$ equalizes $ip$ and $1_{H_B}$. The Yoneda lemma writes $ip=H_b$ for an endomorphism $b:B\to B$. An equalizer $A\to B$ of $b$ and $1_B$ gives the same presheaf equalizer, since the Yoneda embedding preserves categorical limits. Uniqueness of equalizers gives $P\cong H_A$.