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  "body": "\n## Statement\n\nLet $\\mathcal{A}$, $\\mathcal{B}$ and $\\mathcal{C}$ be [[abelian categories|Y9CZ3MTNF1]], and assume that $\\mathcal{A}$ and $\\mathcal{B}$ have enough injectives.\nLet $G \\colon \\mathcal{A} \\to \\mathcal{B}$ and $F \\colon \\mathcal{B} \\to \\mathcal{C}$ be left [[exact|S1G61TDW0K]] [[additive functors|KAN07YC9T2]].\nAssume that $G$ sends every [[injective object|5QRFTBF4PG]] of $\\mathcal{A}$ to an [[$F$-acyclic object|F2CMPBHQ02]] of $\\mathcal{B}$.\nThen for every object $X$ of $\\mathcal{A}$ there is a first quadrant [[spectral sequence|GG8AYV847V]]\n$$E_2^{p,q} = (R^p F)(R^q G)(X) \\Longrightarrow R^{p+q}(F \\circ G)(X),$$\nnatural in $X$, where $R^n$ denotes the right [[derived functors|NPMC3K6J2F]].\n\n## Remarks\n\nTo prove it, take an injective resolution $X \\to I^\\bullet$ and a Cartan–Eilenberg resolution of the complex $G(I^\\bullet)$.\nApply $F$, and compare the two spectral sequences of the resulting double complex.\nBecause $G(I^n)$ is $F$-acyclic, one of them collapses and computes $R^n(F \\circ G)(X)$; the other has the stated $E_2$ page.\nWithout the acyclicity hypothesis the derived functor of $F \\circ G$ is not computed by this double complex, and the statement can fail.\nOne example is the [[Leray spectral sequence|8BMQ90EJPR]], where $G = f_*$ for a continuous map $f$ and $F$ is the global sections functor.\nAnother is the [[Lyndon–Hochschild–Serre spectral sequence|VC3QNG6TKR]], where $G$ and $F$ take invariants under a normal subgroup and under the quotient group.\nThe low degree terms of the spectral sequence give the exact sequence $0 \\to R^1 F(G X) \\to R^1(F G)(X) \\to F(R^1 G(X)) \\to R^2 F(G X) \\to R^2(F G)(X)$.\n\n## Explanations\n\n### L1 human\n\nIf a functor is done in two steps, its derived functors are related to the derived functors of the two steps.\nThe relation is not a simple formula but a spectral sequence: a table of groups that, after several corrections, gives the answer.\nThe one condition is that the first step sends the simple building blocks, the injective objects, to objects that the second step treats as simple.\n\n### L2 human\n\nIn practice the theorem is used in low degrees, through the five-term exact sequence.\nIt is also used when the $E_2$ page vanishes outside one row or column, so that $R^n(F \\circ G)$ can be read off directly.\nIn the language of [[derived categories|5ST7NVJFSM]] it says that $R(F \\circ G) \\cong RF \\circ RG$.\n",
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