ScalingStacks

Remark 1.6. It is important not to confuse the center $\mathcal{Z}(\mathcal{C})$ of a monoidal $\infty$-category $\mathcal{C}$ with the endomorphisms of the identity functor of the underlying $\infty$-category. The former is again an $\infty$-category depending on the monoidal structure of $\mathcal{C}$, while the latter is an algebra with no relation to the monoidal structure of $\mathcal{C}$. For example if $\mathcal{C}=A-\operatorname{mod}$ is modules over an associative of $A_{\infty}$-algebra $A$, then the endomorphisms of the identity of $\mathcal{C}$ are calculated by the (topological) Hochschild cohomology of $A$, not of the $\infty$-category $\mathcal{C}$.