Equalizer of reflexive pair A reflexive pair is a pair of parallel morphisms $f,g:X\to Y$ having a common section, i.e. a map $s:Y\to X$ such that $f\circ s = g\circ s = id_Y$. Reflexive maps are famous because their coequalizers are reflexive coequalizers (see link above). My question is about the _equalizer_ of a reflexive pair: > Is the equalizer of the pair above given by $(Y,s)$? It clearly makes the right diagram commute. But is it universal? Mind that I am _not_ talking about the dual of the reflexive equalizer (i.e. the coreflexive equalizer), I want the pair to be _reflexive_ , not coreflexive.

It is not necessarily universal. For example, if $f$ is a split epimorphism, with $s$ a section, then $f,f$ is a reflexive pair, but unless $f$ and $s$ are isomorphisms, $s$ is not the equalizer of $f$ and $f$, since the equalizer of two equal maps is just the identity.

There is however one notable case where $s$ is the equalizer of $f$ and $g$ : when $f$ and $g$ are jointly monic, so that they form a reflexive _relation_. Then if $f\circ t=g\circ t$, one can show that $t=s\circ f\circ t$. Indeed, it suffices, since $f$ and $g$ are jointly monic, to check that $$f\circ s\circ f\circ t=f\circ t$$ and $$g\circ s\circ f \circ t=f\circ t=g\circ t.$$ Moreover, such a factorisation is clearly unique, since $s$ is a (split) monomorphism.