Suppose I have a coalgebra $\mathcal{C}$ in the $\infty$-category of presentable stable $\infty$-categories (with continuous functors). If the structure functors for $\mathcal{C}$ all have continuous right adjoints, then passing to right adjoints I get an algebra structure on $\mathcal{C}$. However, in the general situation, I only have a full stable subcategory $\mathcal{D}$ of $\mathcal{C}$ such that the right adjoints preserve $\mathcal{D}$, i.e. they induce $\mathcal{D}\otimes \mathcal{D}\rightarrow\mathcal{D}$, and restricting to $\mathcal{D}$ they are continuous. Note that $\mathcal{D}$ is not necessarily a sub-coalgebra of $\mathcal{C}$.

Is there a clean way to show that in the above situation, $\mathcal{D}$ naturally induces an algebra structure? All I can say is that all the homotopical data needed are contained in the coalgebra structure of $\mathcal{C}$, but this seems not so clean.