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.