Epimorphisms of infinite triangular and unitriangular matrices

We discuss the form of all epimorphisms of T∞(F)T∞(F) – the group of all infinite upper triangular matrices over a field, into itself. We show that every such epimorphism is a composition of some standard maps. Moreover, we present an analogous result for UT∞(F)UT∞(F). We use the obtained results to describe the groups of automorphisms of these groups.