the fundamental group of double torus( $T\#T$) is not abelian

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I find this proof from Munkres Topology lacking rigour and a lot of precision.

can someone write down the equations that have been mentioned in the proof and probably provide some feeling or intuition for them?

does the intersection point of the figure eight of $T\#T$ lie on the dotted circle, because if it does then the inverse of $h$ would map back the intersection point of the figure eight in $Y$ to the whole dotted circle and wouldn't be a homeomorphism, right?

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