I think there is a typo in page 133, theorem 4.2.3.

"Proof. Suppose that (g, eta, epsilon) is a quasi-inverse for f ."

In the definition of qinv, the first coordinate corresponds to an homotopy between fg and Id. In the theorem it is called eta. But in the definition of ishae (page 132), eta is taken to be an homotopy between gf and Id. I think that this mismatch would be fixed simply by changing the first sentence of the proof to be "suppose that (g, epsilon, eta) is a quasi-inverse...", to match the definitions of eta and epsilon

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