  If f is a bijection from C^n onto a complex manifold M, which conjugates
every holomorphic map in C^n 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 C^n 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 C^n with isomorphic endomorphism semigroups but which are
neither biholomorphically nor antibiholomorphically equivalent.
