Notes on Several Orthogonal Classes of Flat and $FP$-injective Functors

In this paper, we develop relative homological algebra in the category of functors from finitely presented modules to Abelian groups. More specifically, we introduce the concepts of $\mathfrak{F}$-injective, $\mathfrak{F}$-projective and $\mathfrak{F}$-flat functors. Such functors appear when we study covers and envelopes of functors. The relationships among these functors are investigated and some applications are given.