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In order to study the forward or backward iteration of a holomorphic self-map \(f\) of a complex manifold \(X\), it is natural to search for a semi-conjugacy of \(f\) with some automorphism of a complex manifold. Examples of this approach are given by the Schroeder, Valiron and Abel equation in the unit disc \(D\). Given a holomorphic self-map \(f\) of the ball \(B^q\), we show that it is canonically semi-conjugate to an automorphism (called a canonical model) of a possibly lower dimensional ball \(B^k\), and this semi-conjugacy satisfies a universal property. This approach unifies in a common framework recent works of Bracci, Gentili, Poggi-Corradini, Ostapyuk. This is done performing a time-dependent conjugacy of the autonomous dynamical system defined by \(f\), 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 hyperbolic holomorphic self-map of \(B^q\).