= Left derived functor
{title2=$L_iF(M)=H_i(F(P_\bullet))$}
For an additive right-exact <functor> on module categories, apply it to a <projective resolution> and take <homology>. Lifts of module maps exist by projectivity; any two lifts are <chain homotopic>, giving independence and functoriality. The zeroth derived functor is naturally $F$, positive degrees vanish on projective modules, and the <horseshoe lemma> produces a long exact homology sequence from each short exact sequence. A <Tor functor> is the left derived functor of a <tensor product of modules> functor.
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