  For a proper geodesic metric space $X$, the Morse boundary $\partial_*X$
focuses on the hyperbolic-like directions in the space $X$. It is a
quasi-isometry invariant. That is, a quasi-isometry between two hyperbolic
spaces induces a homeomorphism on their boundaries. In this paper, we
investigate additional structures on the Morse boundary $\partial_*X$ which
determine $X$ up to a quasi-isometry. We prove that, for $X$ and $Y$ proper,
cocompact spaces, a homeomorphism $f$ between their Morse boundaries is induced
by a quasi-isometry if and only if $f$ and $f^{-1}$ are bih\"older, or
quasi-symmetric, or strongly quasi-conformal.
