Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an equivalence
$$\mathcal{D}(B\otimes\_A^LC):=\operatorname{Mod}\_{B\otimes\_A^LC}(\mathrm{Sp})\simeq\mathcal{D}(B)\otimes\_{\mathcal{D}(A)}\mathcal{D}(C)$$
(with tensor products taken appropriately, perhaps in $\operatorname{Cat}\_\infty(\mathcal{K})$ or $\mathrm{Pr}^L\_{st}$)?


I am aware that if we omit the base ring $A,$ this holds true. More precisely, for any symmetric monoidal $\infty$-categories $\mathcal{C}$ and $\mathcal{D}$ and commutative algebra objects $B\in\operatorname{CAlg}(\mathcal{C}),$ $C\in\operatorname{CAlg}(\mathcal{D}),$ the natural map
$$
\operatorname{Mod}\_{B}(\mathcal{C})\otimes\operatorname{Mod}\_C(\mathcal{D})\to\operatorname{Mod}\_{B\otimes C}(\mathcal{C}\otimes\mathcal{D})
$$
is an equivalence (where the tensor products are taken in $\operatorname{Cat}\_\infty(\mathcal{K})$). In the special case $\mathcal{C} = \mathcal{D} = \mathrm{Sp},$ we get
$$
\operatorname{Mod}\_{B}(\mathrm{Sp})\otimes\operatorname{Mod}\_C(\mathrm{Sp})\to\operatorname{Mod}\_{B\otimes C}(\mathrm{Sp}\otimes\mathrm{Sp})\simeq\operatorname{Mod}\_{B\otimes C}(\mathrm{Sp}).
$$
However, I'm not sure how to bootstrap to the relative case from here.


If this is true, I would appreciate a reference or proof, ideally one which is as hands-on as possible so I can understand these objects better. While I want to know if this holds in the generality above, if there's a more intuitive or explicit way to see this at the level of DG-categories, I would also be interested in that.



Does formation of the derived $\infty$-category preserve pushouts?
A hands-on explanation: Relative tensor products like $B\otimes\_AC$ are computed as the colimit of the simplicial object $B\otimes A^{\otimes \bullet} \otimes C$. The functor $\mathsf{Mod}\_{(-)}: \mathsf{Alg}(\mathsf{Sp}) \to \mathsf{Pr}^{L, \mathrm{st}}\_{\mathsf{Sp}/}$ preserves all colimits and is symmetric monoidal, so apply $\mathsf{Mod}\_{(-)}$ everywhere and we get the formula for the relative tensor product of categories of modules.


A 'reference-heavy' explanation:


The reference that $A \mapsto \mathsf{Mod}\_A$ is symmetric monoidal is HA.4.8.5.16. And, the statement about having a right adjoint (when valued in 'presentable stable $\infty$-categories with a distinguished object') is HA.4.8.5.11.


So now we just need a general thing about when $F: \mathcal{C} \to \mathcal{D}$ being a symmetric monoidal left adjoint induces a colimit-preserving functor on $\mathsf{CAlg}(-)$, but that'll be true when the tensor product in $\mathcal{C}$ and $\mathcal{D}$ distribute over colimits- then we get (i) $\mathsf{CAlg}(F)$ preserves coproducts since those are tensor products, and (ii) $\mathsf{CAlg}(F)$ preserves sifted colimits since those are computed on underlying objects (HA.3.2.3.2). Thus $\mathsf{CAlg}(F)$ preserves all colimits, and, in particular, pushouts.


