ScalingStacks

Since \(\mathcal C\) is presentably symmetric monoidal, it follows that centralizers and centers exist [Lur17, Cor. 5.3.1.15] and hence, by applying corollary 7.4.12, that the latter space is equivalent to \[\left(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/Z_1(A)}\right)^{\simeq} \times_{\left(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/A}\right)^{\simeq} } \{\mathrm{id}_A\}.\] Applying lemma 7.4.14 to the \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)\), we find that the functor \[\mathrm{Alg}_{\mathbb E_2}(\mathcal C)_{/Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)/A} \{\mathrm{id}_A\}\rightarrow\mathrm{Alg}_{\mathbb E_2}(\mathcal C) \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)} \{A\}\] is an equivalence. It therefore suffices to show that the forgetful functor \[ \mathrm{Alg}_{\mathbb E_2}(\mathcal C)_{/Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)/A} \{\mathrm{id}_A\} \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/A}} \{\mathrm{id}_A\}\] is an equivalence.