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Let $\mathcal{A}$ and $\mathcal{B}$ be two dg categories, and let $M \in \text{Mod}_{\mathcal{B}}^{\mathcal{A}}$ be a $\mathcal{A}$-$\mathcal{B}$ bimodule. In many references in the framework of dg-categories, the following 'tensor functor' is mentioned $$ - \otimes_{\mathcal{A}}M: \text{Mod}_\mathcal{A} \rightarrow \text{Mod}_\mathcal{B} $$ but none of them give a concrete definition and proof of the Quillen adjunction to $\text{Hom}_{\mathcal{B}}(M,-)$, which makes me confused.

In this paper Relative Calabi-Yau structures, $- \otimes _{\mathcal{A}} M$ is defined as the left Kan extension of $M$ along enriched Yoneda embedding $Y: \mathcal{A} \rightarrow \text{Mod}_{\mathcal{A}}$ without reasons for the existence of this Kan extension.

I want to know more details about the right definition of this functor.

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    $\begingroup$ Aren't these things cocomplete? In this case, the Kan extension will exist. $\endgroup$ – user40276 Sep 1 '17 at 23:15

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