Suppose given an $\mathbb E_n$-monoidal presentable $\infty$-category $\mathcal C$ (wrt. the Lurie tensor product $\otimes$), and $\mathcal D$ a presentable $\infty$-category. Suppose given a pair of functors $A:\mathcal C\to \mathcal D, B:\mathcal D\to \mathcal C$, such that $AB=id_{\mathcal D}$. Under which conditions can we deduce that there exists a canonical $\mathbb E_n$-monoidal structure on $\mathcal D$ itself?
Intuitively, $\mathcal D$ inherits an operation from the following composition (denote by $m_{\mathcal C}$ the operation in $\mathcal C$):
$$\mathcal D\otimes \mathcal D\xrightarrow{B\otimes B}\mathcal C\otimes \mathcal C \xrightarrow{m_{\mathcal C}}\mathcal C\xrightarrow{A}\mathcal D.$$ My question is essentially: under which conditions on $A, B$ do all associativities/coherences transfer to $\mathcal D$?
For example: would it be sufficient to require that the diagram $\require{AMScd}$ \begin{CD} \mathcal D\otimes \mathcal D @>>> \mathcal D\\ @V V V @VV V\\ \mathcal C\otimes \mathcal C @>>> \mathcal C \end{CD}
resulting from the above composition is a pullback?