https://cattheory.subwiki.org/w/index.php?action=history&feed=atom&title=Natural_isomorphismNatural isomorphism - Revision history2026-04-27T01:58:01ZRevision history for this page on the wikiMediaWiki 1.41.2https://cattheory.subwiki.org/w/index.php?title=Natural_isomorphism&diff=58&oldid=prevVipul: New page: {{basicdef}} ==Definition== ===Symbol-free definition=== A '''natural isomorphism''' between functors is a natural transformation between the functors that has a two-sided inverse w...2008-12-09T23:42:14Z<p>New page: {{basicdef}} ==Definition== ===Symbol-free definition=== A '''natural isomorphism''' between functors is a <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformation</a> between the functors that has a two-sided inverse w...</p>
<p><b>New page</b></p><div>{{basicdef}}<br />
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==Definition==<br />
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===Symbol-free definition===<br />
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A '''natural isomorphism''' between functors is a [[natural transformation]] between the functors that has a two-sided inverse which is also a natural transformation.<br />
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===Definition with symbols===<br />
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Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]] and <math>\mathcal{F},\mathcal{G}:\mathcal{C} \to \mathcal{D}</math> are [[functor]]s. A natural isomorphism is a natural transformation <math>\eta: \mathcal{F} \to \mathcal{G}</math> such that there exists a natural transformation <math>\nu:\mathcal{G} \to \mathcal{F}</math> such that the [[composition of natural transformations|composites]] <math>\eta \circ \nu</math> and <math>\nu \circ \eta</math> are both [[identity natural transformation]]s. Specifically, <math>\eta \circ \nu</math> is the identity transformation from <math>\mathcal{G}</math> to itself, and <math>\nu \circ \eta</math> is the identity transformation from <math>\mathcal{F}</math> to itself.</div>Vipul