Since the seminal work of Nambooripad (among others), people have been interested in representing semigroups by categories. This led to the well-known equivalence between inverse semigroups and inductive groupoids (the ESN theorem), or the less-known equivalences between the category of regular semigroups, the category of regular inductive groupoids, and the category of cross-connections. In the talk, we will describe an equivalence for the whole category of semigroups (or monoids), based on small categories with strict factorization systems. Taking natural transformations into account, this extends to an equivalence for a 2-category of unital semigroups. An equivalence of unital semigroups in this 2-category is nothing but Morita equivalence (of monoids).