  We prove the existence and the essential uniqueness of canonical models for
the forward (resp. backward) iteration of a holomorphic self-map $f$ of a
cocompact Kobayashi hyperbolic complex manifold, such as the ball
$\mathbb{B}^q$ or the polydisc $\Delta^q$. This is done performing a
time-dependent conjugacy of the dynamical system $(f^n)$, obtaining in this way
a non-autonomous dynamical system admitting a relatively compact forward (resp.
backward) orbit, and then proving the existence of a natural complex structure
on a suitable quotient of the direct limit (resp. subset of the inverse limit).
As a corollary we prove the existence of a holomorphic solution with values in
the upper half-plane of the Valiron equation for a holomorphic self-map of the
unit ball.
