The notion of a simplification of a homomorphism is introduced and investigated. Its usefulness is demonstrated in providing rather short proofs of the following results: (i) Given an arbitrary homomorphism h and arbitrary words x, y it is decidable whether or not there exists an integer n such that hn(x) = hn(y). (ii) Given an arbitrary homomorphism h and arbitrary words x, y it is decidable whether or not there exists integers n and r such that hn(x) = hr(y). (iii) Given an arbitrary DOL system G and an arbitrary integer d it is decidable whether or not G is locally catenative of depth not larger thand d. (iv) The equivalence problem for elementary polynomially bounded DOL systems is decidable.