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There's clearly a notation for isomorphism. It's just $\cong$. But is there notation to indicate that an isomorphism is natural? And in general, for morphisms which are natural?

I ask because one of my professors complained about the fact that mathematicians haven't developed a notation, and I may remember hearing of such a notation, but I can't remember what it was.

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    $\begingroup$ It's not clear this is necessary. One could write the appropriate functors, and then use $\cong$ to say that the functors are isomorphic. $\endgroup$
    – S. Carnahan ♦
    Commented Nov 11, 2010 at 4:05
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    $\begingroup$ Merely knowing two functors are naturally isomorphic is often inadequate: it is important to know what the isomorphism is (or perhaps its inverse), or at least how it can be uniquely characterized (so it can be used in many ways). $\endgroup$
    – BCnrd
    Commented Nov 11, 2010 at 5:03
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    $\begingroup$ I agree with Scott - it is far better to say that two functors are isomorphic, than just to display an isomorphism and say it is natural. Because, what is it natural with respect to? There are some natural isomorphisms that are natural only with respect to arrows in a subcategory of the domain of the corresponding functor, and this is crucial information. $\endgroup$
    – David Roberts ♦
    Commented Nov 11, 2010 at 5:21
  • $\begingroup$ If you want to know what the isomorphism is, you could write something like $\overset{\simeq}{\underset{\phi}{\longrightarrow}}$. $\endgroup$
    – S. Carnahan ♦
    Commented Nov 11, 2010 at 7:46

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I don't think there's a globally accepted notation. I like Milne's convention who writes $A \approx B$ if the objects $A$ and $B$ are isomorphic in some way and $A \simeq B$ if the objects $A$ and $B$ are naturally isomorphic.

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    $\begingroup$ It's even better to put the tilde over an arrow showing the direction of the map that is an isomorphism. $\endgroup$
    – Peter Samuelson
    Commented Nov 11, 2010 at 7:47

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