Cograde conditions and cotorsion pairs

Let $R$ and $S$ be rings and ${}_R{\omega }_S$ a semidualizing bimodule. We study when the double functor ${\text{Tor}}_i^S(\omega ,{\text{Ext}}_R^i(\omega ,-))$ preserves epimorphisms and the double functor ${\text{Ext}}_R^i(\omega ,{\text{Tor}}_i^S(\omega ,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $\omega$.