  In this work we provide a generalization of the celebrated Myers-Nakai
Theorem for Riemannian manifolds to the framework of non-reversible Finsler
manifolds, based on the ideas used in a previous generalization for reversible
Finsler manifolds proved in \cite{GJR-13}. In the reversible (i.e. metric)
case, the function space used to characterize Finsler isometries is the normed
algebra $C^1_b(M)$, of bounded and $C^1$-smooth real valued functions with
bounded derivative on a Finsler manifold $M$, endowed with its natural norm.
This function space had to be adapted in order for it to reflect the
quasi-metric structure of non-reversible Finsler manifolds, resulting in a
partial loss of the normed space structure. In order to achieve the desired
result, we define new algebraic/quasi-metric structures to model the behavior
of the aforementioned function space. The construction is based on the cone of
smooth semi-Lipschitz functions.
