# Decide the existence of a string homomorphism

Consider the following problem:

Given two strings x,y, decide whether there exists a string homomorphism f such that f(x)=y.

It is easy to show that this problem is in $NP$. Are there other things we can say about this problem? e.g. Is it in $coNP$, or even $P$?

This problem seems very natural, so I am not surprised if it has been studied thoroughly. However I could not find this problem in literature.