We have the following diagram:
We want to prove the existence of such $h$ with $p$ and $q$ strong epimorphisms.
- Why can't we apply directly the definition of strong epimorphism to the epimorphism $p$, the monomorphism $j$ and the lift $(q\circ f , g\circ i)$ ? Wouldn't that immediately give an $h$ that makes everything commute?
Borceux instead does this:
- Do we ever use $q$ is (strong-)epi and $i$ is mono?