I was wondering how to determine what are the regular epimorphisms (i.e. those which are coequalizers) in $\mathrm{hTop}^{\mathrm{connectedCW}}_*$, the category of pointed connected CW complexes modulo homotopy?
My motivation for asking is that (if I'm not mistaken) these should be sent to monomorphisms by $H^j$ for any reduced generalized Eilenberg-Steenrod cohomology $H^{\bullet}$ and integer $j\geq0$, due to e.g. Brown representability.