a$ for some inner automorphism $i_x$.

Feighn-Handel used the following proposition to show the coherence of ascending HNN extensions of free groups; the proposition and the next lemma allow us to rewrite presentations of ascending HNN extension subgroups of ascending HNN extension groups so that fibers of the subgroup lie in fibers of the ambient group.

\[rewrite\] Suppose $\Phi:\mathbb F \to \mathbb F$ is an injective endomorphism of a free group (of possibly infinite rank) and $G = \mathbb F*_\Phi = \left\langle\, \mathbb F, t~|~t^{-1} x t = \Phi(x), \forall x\in\mathbb F \,\right\rangle$. Let $p: G \to \mathbb Z$ be the natural fibration that maps $\mathbb F \mapsto 0$ and $t \mapsto 1$.

If $H \le G$ is finitely generated and $\left.p\right|_H$ is not trivial and not injective, then there is an isomorphism $\iota: F*_A \to H$, natural fibration $\pi: F*_A \to \mathbb Z$, and injective map $n:\mathbb Z \to \mathbb Z$ such that $n(1) \ge 1$ and $\left.p\right|_H \iota = n \pi$, where $F*_A$ is an HNN extension of a finitely generated free group $F$ over a free factor $A \le F$.

The next lemma will be used to show certain subgroups of a mapping torus are also mapping 