E-M factorization system question

I am working on a construction which, given a category with an E-M factorization system (such as onto-oneone functions in the category of sets or universal algebras of a given type) provides an embedding F:A --> B where A is the given category, and F embeds A fully into B. B also has an E-M factorization system. However, if a is an object of A, then the e's of Fa are exactly all filters in the lattice of e's of a in A, and the m's of Fa are exactly all filters in the lattice of m's of a in A. The question, of course, is, has this been done by anyone before and if so, please provide a reference to that work. whr