  Let $\mathbb{A}$ be a $2$-category with suitable opcomma objects and
pushouts. We give a direct proof that, provided that the codensity monad of a
morphism $p$ exists and is preserved by a suitable morphism, the factorization
given by the lax descent object of the higher cokernel of $p$ is up to
isomorphism the same as the semantic factorization of $p$, either one existing
if the other does. The result can be seen as a counterpart account to the
celebrated B\'{e}nabou-Roubaud theorem. This leads in particular to a
monadicity theorem, since it characterizes monadicity via descent. It should be
noted that all the conditions on the codensity monad of $p$ trivially hold
whenever $p$ has a left adjoint and, hence, in this case, we find monadicity to
be a $2$-dimensional exact condition on $p$, namely, to be an effective
faithful morphism of the $2$-category $\mathbb{A} $.
