Foxby equivalence relative to $C$-$f{p}_n$-injective and $C$-$f{p}_n$-flat modules

Let $R$ and $S$ be rings, $C={}_S{C}_R$ a (faithfully) semidualizing bimodule, and $n$ a positive integer or $n=\mathrm{\infty }$. In this paper, we introduce the concepts of $C$-$f{p}_n$-injective $R$-modules and $C$-$f{p}_n$-flat $S$-modules as a common generalization of some known modules such as $C$-$F{P}_n$-injective (resp. $C$-weak injective) $R$-modules and $C$-$F{P}_n$-flat (resp. $C$-weak flat) $S$-modules. Then we investigate $C$-$f{p}_n$-injective and $C$-$f{p}_n$-flat dimensions of modules, where the classes of these modules, namely $Cf{p}_{n}I(R{)}_{\le k}$ and $Cf{p}_{n}F(S{)}_{\le k}$, respectively. We study Foxby equivalence relative to these classes, and also the existence of $Cf{p}_{n}I(R{)}_{\le k}$ and $Cf{p}_{n}F(S{)}_{\le k}$ preenvelopes and covers. Finally, we study the exchange properties of these classes, as well as preenvelopes (resp. precovers) and Foxby equivalence, under almost excellent extensions of rings.