Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to \mathcal A'$ be a dg-functor. This clearly induces a dg-functor $F' :=F \otimes 1_{\mathcal B} : \mathcal A \otimes \mathcal B \to \mathcal A' \otimes \mathcal B$, which is fully faithful if $F$ is such.

A $\mathcal A$-$\mathcal B$-bimodule $G$ ($\mathcal A$ acting on the left, $\mathcal B$ acting on the right) is by definition a right $\mathcal A^{\text{op}} \otimes \mathcal B$-dg-module, that is, a dg-functor $\mathcal A^{\text{op}} \otimes \mathcal B \to \mathbf{C}_{\text{dg}}(k)$, where $\mathbf{C}_{\text{dg}}(k)$ is the dg-category of cochain complexes of $k$-modules. It is well-known (see this, for example) that dg-modules can be extended along dg-functors. My question is the following. Let $G$ be a $\mathcal A$-$\mathcal B$-bimodule, and let $\mathrm{Ind}_{F'}(G)$ its extension along $F'$. Now, assume that $G$ is a quasi-functor, that is, $G(A,-)$ is quasi-isomorphic to a representable right $\mathcal B$-dg-module for all $A \in \mathcal A$. Then, is $\mathrm{Ind}_{F'}(G)$ again a quasi-functor? If needed, assume that $F$ is fully faithful.