  We give an alternative criteria for when a pair of Bourn-normal monomorphisms
Huq-commute in a unital category. We use this to prove that in a unital
category, in which a morphism is a monomorphism if and only if its kernel is
zero morphism, a pair of Bourn-normal monomorphisms with the same codomain
Huq-commute as soon as they have trivial pullback. As corollaries we show that
several facts known only in the protomodular context are in fact true in more
general contexts.
