The immediate purpose of this paper is to show how Li's characterization of Green's relations on monoids of strong endomorphisms of graphs [Li Weimin, Green's relations on the strong endomorphism monoid of a graph, Semigroup Forum 47 (1993), 209--214] is related to earlier work of J. Klasa [Semisimplicity and von Neumann's regularity, Semigroup Forum 2 (1971), 354--361] on categories. Li's results are thus extended to some categories of strong graph homomorphisms with more than one object, and a finiteness requirement is weakened. A further purpose is to extend Klasa's results in such a way as to better reflect the factorization properties of interesting concrete categories.