  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.
