Representable retract criterion
= Representable retract criterion
Suppose a <small category> has <equalizers>. If a <categorical presheaf> $P$ is a retract $P\xrightarrow{i}H_B\xrightarrow{p}P$, 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 $P\cong H_A$.