Let \(R\) be a ring and let \(T\) be an additive covariant functor from \(R\)-modules to abelian groups. For an \(R\)-module \(A\), choose a projective resolution \(\cdots\to P_1\to P_0\to A\to 0\) and let \(P_\bullet\) be the complex \(\cdots \to P_1\to P_0\to 0\) with \(A\) deleted. The left derived functors of \(T\) are \[ L_n T(A) \coloneqq H_n\qty{T(P_\bullet)}, \qquad n\geq 0 .\]
Any two projective resolutions of \(A\) are chain homotopy equivalent, so \(L_nT(A)\) is well defined up to natural isomorphism, and a morphism \(A\to A'\) induces \(L_nT(A)\to L_nT(A')\). If \(T\) is right exact, then \(L_0T\cong T\).
[@Wei94].