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Let (F, G) be an adjoint pair from a category 4 to a category B. The contravariantly finiteness of subcategories is proved to be preserved by F, and the covariantly finiteness of subcategories is preserved by F under some additional conditions. If A and B are length categories, G(Im (F)) = Im (G), and F is full,then the inclusion i: Im (F) → B has a right adjoint. This gives a partial answer to a question posed by Auslander and Reiten in [2].