Regular monics are not closed under composition, neither are regular epics. Adámeck, Herrlich, and Strecker provide examples of each. For regular monics, they use the category FHaus of functionally Hausdorff spaces (sec. 7J). For regular epics, their example lives in Cat.
It seems that regular epics are closed under composition in FHaus. (The proof that regular epics in Top coincide with quotient maps can be adapted to FHaus, and quotient maps are obviously closed under composition.) Is there a topological example of the failure of regular epics to be closed under composition?
Preferably I would like one in a full subcatgory of Top, and where the class of spaces has a "not too contrived" definition. FHaus is "not too contrived", by my lights.
Failing an example in a subcategory of Top, is there an example in hTop or a subcategory of it?