Given a morphism of schemes which is flat, proper, and "fiber-by-fiber semi-stable'', we study the problem of extending the morphism to a morphism of fine log schemes, which is log smooth, integral, and vertical. The problem is rephrased in terms of a functor on the category of fine log schemes over the base, and the main result of the paper is that this functor is representable by a fine log scheme whose underlying scheme maps naturally to the base by a monomorphism of finite type. In the course of the proof, we also generalize results of Kato on the existence of log structures of embedding and semi-stable type.