Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top dimensional simplices of $A(S)$ the map $\lambda$ is induced by a homeomorphism $\Phi$ of $S$.
Question: How to show that $\lambda$ is induced by a homeomorphism of $S$ in the whole of $A(S)$?
P.S: In the context I have heard about it the speaker said it followed from "Mosher's connectivity argument." I searched about it but I didn't find any reference or answer.