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From this post of Ravi Vakil’s blog:

 

I’ve gradually come around to the idea that when learning about some category for the first time, the notion of isomorphism is pedagogically prior to the notion of morphism. I won’t argue about whether it is logically prior; that’s not my point. When learning groups, students first propose the notion of isomorphism (as they figure out what they mean by the intuition of two groups being the same) before the notion of how you map from one group to another. With schemes too isomorphisms come first.

 

In mathematics notation, we have a symbol for “is isomorphic to” (\cong). But this is the wrong notion in general: we need a symbol for “a map that is an isomorphism”. We already have a reasonable answer: a right arrow with a \sim on top. But might it be nicer to make it look slightly different, and have a symbol that looks like a \cong, where the bottom is a rightarrow? I like this idea because although it is (slightly) new notation, it is patently clear what it means, and also useful.

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