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How is the canonical isomorphism defined here ?

$Y\times_{Y\times_Z Y}X\times_Z X\overset{\sim}\longrightarrow X\times_Y X$

additionally given are the maps $f:X\to Y$ and $g:Y\to Z$ both being separated morphisms. I don’t know whether this is necessary or not

here it is line $8$ from top

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Jno
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  • Some context would be appreciated, but these sorts of things usually follow from the universal properties (and therefore hold in any category). – Pol van Hoften Feb 21 '20 at 17:28
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    user45878 this comes up in the proof of a theorem where $f$ and $g$ are separated morphisms as above and the aim is to show their composition is also separated. – Jno Feb 21 '20 at 17:33
  • I assume you're talking about Proposition 10.1.13. of Vakil, where he refers to exercise 1.3.S, which is purely an exercise in category theory. For this, see https://mathoverflow.net/questions/80797/magic-square-of-fibered-products-vague-unclear – Pol van Hoften Feb 21 '20 at 17:42
  • Is it the same as in this question ? – Arnaud D. Feb 21 '20 at 18:06

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