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Fix a (possibly noncommutative) ring R and consider the category $Rings_R$ of rings $S$ with a map $S \to R$. There is a extension of scalars functor to R-bimodules $Rings_R \to R-Mod-R$ which sends $S \mapsto R\otimes_S R$.

This functor turns out to preserve colimits. Does it have a right adjoint?

More specifically, I am interested in showing that its derived version $S \mapsto R \otimes^L_S R$ preserves homotopy colimits, where $S$ and $R$ are now dg categories. Does this functor have a derived right adjoint?

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    $\begingroup$ It doesn't preserve initial objects, since surely $\mathbb{Z} \to R$ is initial in $Rings_R$, but the initial bimodule is the zero bimodule. $\endgroup$
    – Todd Trimble ♦
    Nov 19, 2017 at 4:05

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