Complete cotorsion pairs in exact categories

We show a cotorsion pair cogenerated by a class is complete under suitable conditions in an arbitrary exact category using the generalized small object argument given by Chorny. This recovers Saor'in and \v{S}\v{t}ov'{i}\v{c}ek’s criterion of the completeness of cotorsion pairs in their efficient exact categories. Examples in the categories of chain complexes of exact categories are given.