Equivalences for the (2-)categories of monoids and unital semigroups

We construct a category equivalent to the category $\mathbf{M}\mathbf{o}\mathbf{n}$ of monoids and monoid homomorphisms, based on categories with strict factorization systems. This equivalence is then extended to the category $\mathbf{M}\mathbf{o}{\mathbf{n}}_{\mathbf{s}}$ of unital semigroups and semigroup homomorphisms. By introducing suitable natural transformations, we turn these equivalences into 2-equivalences between 2-categories. The 2-category $\mathbf{M}\mathbf{o}{\mathbf{n}}_{\mathbf{s}}^{\mathbf{2}}$ constructed this way proves the good one to study Morita equivalence of monoids.