This amounts to saying that a string w ∈ Σ * is a path in D if and only if there is a path w in D that satisfies the following: w has a prefix of the form σ 0,1 σ 1,1 , (b) w has a suffix of the form σ 1,1 σ 1,0 , and (c) w is obtained from w by applying the homomorphism that replaces each σ 1,1 by σ and erases all other symbols.
because the path language of [[M ]] is regular, so is that of [[M]], because it is obtained from the former by intersection with two regular languages and applying a homomorphism.