  Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We study
when the double functor $\Tor^S_i(\omega, \Ext^i_{R}(\omega,-))$ preserves
epimorphisms and the double functor $\Ext_{R}^i(\omega, \Tor_i^{S}(\omega,-))$
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 $\omega$.
