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We construct a functor from the category of modules over the trigonometric (resp., rational) Cherednik algebra of type gl(l) to the category of integrable modules of level l over a Yangian for the loop algebra sl(n) (resp., over a subalgebra of this loop Yangian) and we establish that it is an equivalence of categories if l+2 < n. Finally, we treat the case of the rational Cherednik algebras of type A(l-1).