# “There is a natural [way/map/etc.]…” [duplicate]

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What is a natural isomorphism?

I have often encountered the phrase "There is a natural [way/map/etc.]..." when describing say isomorphisms, maps, etc. What exactly does "natural" mean?

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## marked as duplicate by joriki, Asaf Karagila, Ｊ. Ｍ., t.b., Willie WongSep 13 '11 at 15:58

You can get a good understanding by replacing the word 'natural' with 'coordinate-free' whenever you see it. A nice example of a 'natural' isomorphism is the demonstration that every vector space $V$ over field $F$ is isomorphic to its double dual, via the mapping $\phi:V\to V^{**}$ defined by $\phi(v)(\alpha) = \alpha(v)$ for each $v\in V$, $\alpha\in V^*$. –  Chris Taylor Sep 13 '11 at 13:12
@Chris: ... given certain restrictions on $V$ (finite dimension for algebraic duals, something more involved for continuous duals). –  Henning Makholm Sep 13 '11 at 14:50