Definition 5.10 (Reflexive / split coequaliser diagram / $G$-split).

• (a)
  We say a parallel pair $A\underset{g}{\overset{f}{\rightrightarrows }}B$ is reflexive if there exists $r:B\to A$ with $fr=gr=1_B$. Note that $FGFA\underset{{𝜀}_{FA}}{\overset{F\alpha }{\rightrightarrows }}FA$ is reflexive, with common right inverse $FA\stackrel{F{\eta }_A}{\to }FGFA$.
• (b)
  By a split coequaliser diagram, we mean a diagram

  

  satisfying $hf=hg$, $hs=1_C$, $gt=1_B$ and $ft=sh$. If these hold, then $h$ is a coequaliser of $(f,g)$ since if $B\stackrel{k}{\to }D$ satisfies $kf=kg$ then $k=kgt=kft=ksh$, so $k$ factors through $h$, and the factorisation is unique since $h$ is (split) epic. Note that any functor preserves split coequalisers.
• (c)
  Given $G:D\to C$, we say a pair $A\underset{g}{\overset{f}{\rightrightarrows }}B$ in $D$ is $G$-split if there’s a split coequaliser diagram

  

  in $C$. The pair $(F\alpha ,{𝜀}_{FA})$ in Lemma 5.9 is $G$-split, since

  

  is a split coequaliser diagram in $C$.