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I find it hard to understand part of Hatcher's proof of exactness of the homotopy group sequence on this page.

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In the fourth line from the bottom, he says "We can tack F onto f", but I don't quite understand this, could you please explain this in a more formal language to me?

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    It is depicted by the supplied image.2017-02-01
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    @Hurkyl I do notice the picture, but I don't know what is the preimage of $f$.2017-02-01

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