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added pointer to:
Thanks for pointing this out.
I have added (here) a more precise pointer to the relevant reverse pasting law for $\infty$-pullbacks, which applies since the point inclusion into a delooping is an effective epi.
On the other hand, we may understand the situation the other way around: First $G/H$ is defined by the second pullback, and then the ordinary $\infty$-pasting law gives the former one.
Finally, I have added pointer to Ex 3.2.35 on p. 104 here for more.
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