Remark 1.4.23. Note that \(\Iso (C)\) encodes separate left and right inverses \(g\) and \(h\) of \(f\), rather than a single two-sided inverse \(f^{-1}\). The reason for this is to guarantee that the fibers of \(\pi _{\Iso }\colon \Iso (C) \to \Ar (C)\) are either empty or contractible, so that being an isomorphism is really a property of a morphism rather than additional structure. This is made precise by Proposition 1.5.6.
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