Maps conjugating holomorphic maps in Cⁿ

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If f is a bijection from Cⁿ onto a complex manifold M, which conjugates every holomorphic map in Cⁿ to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorphic endomorphisms of Cⁿ to the semigroup of holomorphic endomorphisms in M, or an epimorphism in the opposite direction for a doubly-transitive M, then it is given by conjugation by some biholomorphic or antibiholomorphic map. We show also that there are two unbounded domains in Cⁿ with isomorphic endomorphism semigroups but which are neither biholomorphically nor antibiholomorphically equivalent.