In Example 5.1.2.4 of Higher Algebra, Lurie explains how there is a bijection between equivalence classes braided monoidal structures on one-category $\mathcal{C}$ and $\mathbf{E}_2$-algebra structures on the nerve $\text{N}(\mathcal{C})$. Likewise for $\mathbf{E}_n$, $n\ge 0$.
Pre-question: how does one interpret this as an equivalence of categories, e.g. is the stronger thing true $$\text{N}_2(\textbf{E}_n\text{-Alg}(\text{1-Cat}))\ \stackrel{?}{\simeq}\ \textbf{E}_n\text{-Alg}(\text{N}_2(\text{1-Cat}))\ =\ \textbf{E}_n\text{-Alg}(\textbf{1-Cat})?$$ Here where $\text{N}_2$ is the nerve of the 2-category $\text{1-Cat}$, $\textbf{1-Cat}\subseteq \text{Cat}_\infty$, and (I think?) $\text{N}_2(\text{1-Cat}) \stackrel{}{=} \textbf{1-Cat}$.
My question is then: if we replace $\text{1-Cat}$ with an arbitrary 2-category $\mathcal{E}$, is the above pre-question (or a salvage thereof) still true?
