  We study the smoothness and preserving orientation properties of a global and
nonautonomous version of the Hartman--Grobman Theorem when the linear system
has a nonuniform contraction on the half line. The nonuniform contraction
implies the existence of a density function (\emph{i.e} a dual type of Lyapunov
function) for the linear system which combined with the above diffeomorphism
allow us to construct a density function for the nonlinear system.
