Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.
In a pushout-pullback diagram, which consists of four morphisms f ： A → B, g ： A → C, α ： C → D and β ： B → D, we give some relations among the covers of these four modules. If ker f∈I（ ）, then g ： A → C is -covering if and only if ： B → D is -covering. If every module has an -precover and kerf E I（ ）, then A and C have isomorphic -precovers if and only if B and D have isomorphic -precovers.
