An extension of the regular completion (Q1850097)

\textit{A. Carboni} [J. Pure Appl. Algebra 103, 117--148 (1995; Zbl 0839.18002)] presented the 2-category of regular categories and functors as the 2-category of algebras of a KZ-doctrine on \(lex\) (the 2-category of finitely complete categories and left exact functors). In order to extend this result, the authors introduce the concept of kernels and kernel arrows in a general category. Then they define a KZ-doctrine on the 2-category of all categories and those functors preserving kernel arrows (and all natural transformations). Its 2-category of algebras is shown to be the (regular epi, mono)-factorizable categories (i.e., those in which every morphism factorizes as a regular epi followed by a mono), with the functors preserving these factorizations (and all natural transformations).