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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.\nConsider a [[commutative diagram|WV898P4X16]]\n$$\\begin{array}{ccccccccc} A \u0026 \\to \u0026 B \u0026 \\to \u0026 C \u0026 \\to \u0026 D \u0026 \\to \u0026 E \\\\ \\downarrow{\\scriptstyle l} \u0026 \u0026 \\downarrow{\\scriptstyle m} \u0026 \u0026 \\downarrow{\\scriptstyle n} \u0026 \u0026 \\downarrow{\\scriptstyle p} \u0026 \u0026 \\downarrow{\\scriptstyle q} \\\\ A' \u0026 \\to \u0026 B' \u0026 \\to \u0026 C' \u0026 \\to \u0026 D' \u0026 \\to \u0026 E' \\end{array}$$\nin $\\mathcal{A}$ whose two rows are [[exact|JB7ZBFBA9C]].\nSuppose that $m$ and $p$ are isomorphisms, that $l$ is an [[epimorphism|RVXDSY04KG]] and that $q$ is a [[monomorphism|Y2NP1JFR86]].\nThen $n$ is an isomorphism.\n\n## Remarks\n\nThe lemma is the conjunction of the two halves of the [[four lemma|JMCKACGCQ0]].\nThe left four columns, with $m, p$ monomorphisms and $l$ an epimorphism, show that $n$ is a monomorphism.\nThe right four columns, with $m, p$ epimorphisms and $q$ a monomorphism, show that $n$ is an epimorphism.\nEach half is a diagram chase.\nA typical use: a map between two short exact sequences that is an isomorphism on the outer terms is an isomorphism on the middle term, by taking $A = A' = E = E' = 0$.\nThe existence of the map $n$ making the diagram commute is essential: $0 \\to \\Z/2\\Z \\to \\Z/4\\Z \\to \\Z/2\\Z \\to 0$ and $0 \\to \\Z/2\\Z \\to \\Z/2\\Z \\oplus \\Z/2\\Z \\to \\Z/2\\Z \\to 0$ have isomorphic outer terms, but $\\Z/4\\Z \\not\\cong \\Z/2\\Z \\oplus \\Z/2\\Z$.\n\n## Explanations\n\n### L1 human\n\nPicture two exact rows of five objects, with vertical maps between them.\nIf the four outer vertical maps are isomorphisms, then so is the middle one.\nIn fact less is needed at the two ends: the far left map only needs to be onto, and the far right map only needs to be one-to-one.\n\n### L2 human\n\nThe five lemma is used constantly to show that a natural map is an isomorphism.\nOne fits the map into a ladder of long exact sequences, for example in homology, where the other maps are already known to be isomorphisms.\nThis is how one proves that a map of spaces that induces isomorphisms on the homology of pieces induces an isomorphism on the whole, as in Mayer–Vietoris arguments.\nIt also proves the splitting lemma and the invariance of derived functors under the choice of resolution.\n",
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