Remark 5.3.21. Observe that there are equivalences \(\CMon (C_*) \simeq \CMon (C)_*\) and \(\CGrp (C_*) \simeq \CGrp (C)_*\). Since \(\CMon (C)\) and \(\CGrp (C)\) are pointed, it follows from Lemma 4.1.9 that the forgetful functor \(C_* \to C\) induces equivalences \[ \CMon (C_*) \iso \CMon (C) \qquadtext { and } \CGrp (C_*) \iso \CGrp (C). \] Using a similar argument we find equivalences \[ \Mon (C_*) \iso \Mon (C) \qquadtext { and } \Grp (C_*) \iso \Grp (C). \]
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