However, for our purpose the following simplified approach is sufficient. A regular mono is a map which can be written as the equalizer of some pair of maps. An epic strong mono is an isomorphism. Any composite of strong monos is a strong mono. Thus f is not a strong mono.
Y are two strong monos.
We have to prove that w is an isomorphism. Then u is an isomorphism as f is a strong mono. But g is a strong mono and v is a pullback of g . So v is also an isomorphism. Hence g as an intersection of isomorphisms is an isomorphism.