Proposition 7.4.6. ([Lur17, Prop. 5.3.1.8]).
Let \(M\) be an object in a monoidal \(\infty\)-category \(\mathcal C\). If \(\mathrm{id}_M\colon M \rightarrow M\) has a centralizer in \(\mathcal C\), then \(M\) has a center. Furthermore, an \(\mathbb E_1\)-algebra \(A\) with a left-action on \(M\) is a center of \(M\) if and only if the underlying maps \(I \rightarrow A\) and \(A\otimes M \rightarrow M\) exhibit \(A\) as a centralizer of \(\mathrm{id}_M\).