Given a homomorphism $h: \Sigma \rightarrow \Delta^*$ such that e.g. $\forall a \in \Sigma: h(a) = \delta$, where $\delta \in \Delta$ (i.e. all symbols from the alphabet $\Sigma$ have the same image $\delta$), is the inverse homomorphism $h^{-1}$ a homomorphism?
Since a homomorphism maps every symbol of the input alphabet to one word from the output alphabet, the inverse transformation to $h$ is a substitution $\sigma: \delta \mapsto \Sigma$ (which maps one symbol to a whole language - alphabet $\Sigma$).
What would be an inverse homomorphism to $h': a \mapsto \epsilon, \forall a \in \Sigma$ or would such a transformation even exist?
EDIT: The question should have rather stated: "Is there always an inverse homomorphism?" There obviously isn't since only isomorphisms have their inverse homomorphisms.