[05ZJ] Proof. By 5.2.2, the ∞\infty-operad End𝒞∗red(X)\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right) is either 𝔼0\mathbb{E}_{0} or 𝔼∞\mathbb{E}_{\infty}. On the other hand, the existence of an HH-structure is equivalent to End𝒞∗red(X)(2)≠∅\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(2\right)\neq\varnothing. Thus, XX admits an HH-structure if and only if End𝒞∗red(X)≃𝔼∞\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)\simeq\mathbb{E}_{\infty} if and only if XX admits a unique ∞\infty-delooping. ∎