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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \\colon X \\to Z$ and $g \\colon Y \\to Z$ be [[morphisms|859F0TFVA3]] of $\\mathcal{C}$.\nA *pullback* of $f$ and $g$ is an object $P$ with morphisms $p_1 \\colon P \\to X$ and $p_2 \\colon P \\to Y$ such that:\n\n1. $f \\circ p_1 = g \\circ p_2$;\n2. for every object $W$ and all morphisms $q_1 \\colon W \\to X$ and $q_2 \\colon W \\to Y$ with $f \\circ q_1 = g \\circ q_2$, there is exactly one morphism $u \\colon W \\to P$ with $p_1 \\circ u = q_1$ and $p_2 \\circ u = q_2$.\n\nThe object $P$ is written $X \\times_Z Y$ and called the *fibre product* of $X$ and $Y$ over $Z$.\n\n## Remarks\n\nA pullback is the [[limit|CHS5AFJGWE]] of the diagram $X \\xrightarrow{f} Z \\xleftarrow{g} Y$, so it is unique up to unique isomorphism.\nIf the [[product|S5744QZV4N]] $X \\times Y$ exists, the pullback is the [[equalizer|CTJ5AD8ST5]] of $f \\circ \\pi_1$ and $g \\circ \\pi_2 \\colon X \\times Y \\to Z$.\n\nExample: in the category of sets, let $P = \\set{(x, y) \\in X \\times Y : f(x) = g(y)}$ with the two coordinate maps.\nClause 1 holds by the choice of $P$.\nGiven $q_1, q_2$ with $f \\circ q_1 = g \\circ q_2$, the map $u(w) = (q_1(w), q_2(w))$ lands in $P$ and is the only map with the required coordinates.\nWhen $g$ is the inclusion of a subset $Y \\subseteq Z$, the pullback is the [[preimage|01AHG4G9RT]] $f^{-1}(Y)$ with its inclusion into $X$.\n\nNon-example: in the category of sets, let $X = Y = \\{0, 1\\}$, $Z = \\{*\\}$ and let $f, g$ be the constant maps.\nThe set $\\{0, 1\\}$ with $p_1 = p_2 = \\id$ satisfies clause 1, but the pair $q_1 = \\id$, $q_2 = (0 \\mapsto 1, 1 \\mapsto 0)$ on $W = \\{0, 1\\}$ admits no $u$ with $p_1 \\circ u = q_1$ and $p_2 \\circ u = q_2$, so clause 2 fails.\n",
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