= Reflexive-pair groupoid formula in a preadditive category
{title2=$b\circ a=a+b-rga,\quad a^{-1}=rfa+rga-a$}
For a <reflexive pair> $f,g:A\rightrightarrows B$ with section $r$ in a <preadditive category>, each hom-set diagram defines a <groupoid>. Arrows $a,b:C\to A$ compose when $ga=fb$, by $b\circ a=a+b-rga$. Identity at $x:C\to B$ is $rx$, and the inverse of $a$ is $rfa+rga-a$. Bilinearity proves the endpoint, identity and associativity laws and compatibility with precomposition in $C$. In the hom-set formulation this does not assume that the composable-arrow pullback exists.
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