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  "body": "\n## Statement\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]], and consider a [[commutative diagram|WV898P4X16]] of two squares side by side:\n$$\\begin{array}{ccccc} A \u0026 \\xrightarrow{a} \u0026 B \u0026 \\xrightarrow{b} \u0026 C \\\\ \\downarrow{\\scriptstyle u} \u0026 \u0026 \\downarrow{\\scriptstyle v} \u0026 \u0026 \\downarrow{\\scriptstyle w} \\\\ D \u0026 \\xrightarrow{d} \u0026 E \u0026 \\xrightarrow{e} \u0026 F \\end{array}$$\nso that $v \\circ a = d \\circ u$ and $w \\circ b = e \\circ v$.\nAssume that the right square is a [[pullback|Q05K04QJ71]].\nThe outer rectangle has top side $b \\circ a$, bottom side $e \\circ d$ and vertical sides $u$ and $w$.\nThen the left square is a pullback if and only if the outer rectangle is a pullback.\n\n## Remarks\n\nBoth directions are checked with the universal property.\nA cone over the outer rectangle gives, through the right pullback, a cone over the left square, and back again.\nThe uniqueness parts match because the right square is a pullback.\nIn the category of sets, a pullback along an inclusion is a preimage.\nThe lemma then says that the preimage under $e \\circ d$ is the preimage under $d$ of the preimage under $e$.\n\nThe hypothesis on the right square is needed.\nLet $A$, $C$, $D$, $E$ and $F$ be one-point sets, let $B$ have two points, and let $a$ pick one of them.\nThe outer rectangle is then a pullback, but the left square is not, because the pullback of $B \\to E \\leftarrow D$ has two points while $A$ has one.\n\n## Explanations\n\n### L1 human\n\nPullbacks can be stacked.\nIf you glue a pullback square to a second square, the big rectangle is a pullback exactly when the second square is.\nIt means that pulling back in two steps is the same as pulling back once along the composite map.\n\n### L2 human\n\nThe lemma is the main tool for computing with pullbacks in diagrams.\nIt shows, for example, that pullback along $g \\circ f$ is the same as pullback along $g$ followed by pullback along $f$, up to isomorphism.\nIn algebraic geometry it gives the transitivity of base change, $(X \\times_S S') \\times_{S'} S'' \\cong X \\times_S S''$.\nThe dual statement holds for pushouts: if the left square is a pushout, then the right square is a pushout if and only if the outer rectangle is.\n",
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    "name": "Pullback pasting lemma",
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        "url": "https://en.wikipedia.org/wiki/Pullback_(category_theory)"
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  "msc": [
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  "name": "Pullback pasting lemma",
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  "url": "https://tamnd.github.io/math2/wiki/pullback-pasting-lemma/",
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