Proof.
Since \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C) \rightarrow\mathcal C\) is conservative, the \(\infty\)-categories in ([00F6]) are \(\infty\)-groupoids, and equivalently given by the fibers of the respective maps of maximal subgroupoids.
For an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal C\), it follows from the free-forgetful adjunction 
and initiality of the unit of \(\mathcal C\) that the free \(A\)-module \(_{A}A\) is initial in \(\mathrm{LMod}_A(\mathcal C)\). This induces a functor \(\mathrm{LMod}_A(\mathcal C) \simeq \mathrm{LMod}_{A}(\mathcal C)_{{}_{A}A/} \rightarrow\mathcal C_{A/}\) which sends a left \(A\)-module \(M\) to the morphism \[\mathrm{act}_1 \colon A\simeq A \otimes I \xrightarrow{\mathrm{id}_A \otimes !} A \otimes M \xrightarrow{\mathrm{act}} M.\]
Applying this functor \(\mathrm{act}_1 \colon \mathrm{LMod}_A(\mathcal C) \rightarrow\mathcal C_{A/}\) fiberwise induces a commuting diagram of spaces
By the universal property of the center, the \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/\mathfrak{Z}(X)}\) is equivalent to the \(\infty\)-category \(\mathrm{LMod}(\mathcal C) \times_{\mathcal C} \{X\}\), and hence the space \(\underline{\mathrm{Alg}}_{\mathbb E_1}(\mathcal C)_{/\mathfrak{Z}(X)} \times_{\mathcal C_{/X}} \{ \mathrm{id}_X\}\) is equivalent to the fiber of the top horizontal map in ([00F8]) at \(\{ \mathrm{id}_X\} \in \mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)\). The functor ([00F6]) is the induced map from the fiber of the top horizontal map of ([00F8]) at \(\mathrm{id}_X\) to the fiber of the bottom horizontal map of ([00F8]) at \(X\).
Let \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\) denote the full subspace of \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)\) on the invertible arrows in \(\mathcal C\). In particular, the composite \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\rightarrow\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C) \xrightarrow{s}\mathcal C^{\simeq}\) is an equivalence. We now show that the composite map of spaces \[ \mathrm{LMod}(\mathcal C)^{\simeq} \times_{\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)} \mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}} \rightarrow\mathrm{LMod}(\mathcal C)^{\simeq} \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal C)^{\simeq}\] is an equivalence, which concludes the proof as \(\{ \mathrm{id}_X\} \in \mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)\) is an object of the full subspace \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\).
It suffices to verify that all fibers of ([00F9]) are contractible. By definition, for \(A\in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\) the fiber is the full subspace of \(\mathrm{LMod}_A(\mathcal C)^{\simeq} \simeq \left(\mathrm{LMod}_A(\mathcal C)_{{}_{A}A/}\right)^{\simeq}\) on those modules \(M\) for which the induced map \(\mathrm{act}_1 \colon A \rightarrow M\) is an equivalence. But since \(\mathrm{LMod}_A(\mathcal C) \rightarrow\mathcal C\) is conservative, this is the full subcategory \(\mathrm{LMod}_A(\mathcal C)_{{}_{A}A/^{\mathrm{iso}}}\) on the invertible module functors \({}_{A}A \rightarrow_{A}M\) and hence contractible. ◻