I employ the vast majority of the post to develop the notion of *quasi-functor* between dg-categories: I think it is important to get the idea.

Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ be the category of cochain complexes of $k$-vector spaces and chain maps. $\mathcal V$ is a symmetric closed monoidal category, and also has a structure of model category (the weak equivalences are the quasi-isomorphisms, the fibrations are the surjective chain maps). Categories enriched over $\mathcal V$ are called dg-categories. $\mathcal V$ is also enriched over itself.

Let $\mathcal A, \mathcal B$ be categories enriched over $\mathcal V$. The category $\mathrm{Fun}_{\mathcal V} (\mathcal A, \mathcal B)$ of $\mathcal V$-functors and $\mathcal V$-natural transformations is also enriched over $\mathcal V$. The (enriched) Yoneda lemma holds, and we have the (fully faithful) Yoneda embedding: \begin{equation} \mathcal A \hookrightarrow \textrm{mod-}\mathcal A =: \mathrm{Fun}_{\mathcal V}(\mathcal A^{\mathrm{op}},\mathcal V). \end{equation}

The category $\textrm{mod-}\mathcal A$ of (right) $\mathcal A$-modules inherits a (levelwise) model structure from $\mathcal V$: for example, quasi-equivalences are given by levelwise quasi-equivalences.

Now, let me give the following definition: given $\mathcal A, \mathcal B$ two $\mathcal V$-categories, a $\mathcal V$-functor $F: \mathcal A \to \textrm{mod-}\mathcal B$ is called a *quasi-functor* if for any $A \in \mathcal A$, $F(A)$ is isomorphic to a representable right $\mathcal B$-module in the homotopy category $\mathrm{Ho}(\text{mod-}\mathcal B)$.

Now, observe that we can define a "tensor product" $\mathcal A \otimes \mathcal B$ of $\mathcal V$-categories. Moreover, $\mathcal V$-functors $\mathcal A \otimes \mathcal B^{\mathrm{op}} \to \mathcal V$ correspond exactly to $\mathcal V$-functors $\mathcal A \to \textrm{mod-}\mathcal B$. They are precisely the *bimodules* (or *profunctors*). Let me denote by $\mathcal A\text{-mod-}\mathcal B$ the $\mathcal V$-category of bimodules (covariant in $\mathcal A$, contravariant in $\mathcal B$). Since it is a category of modules, it is a model category, with the levelwise structure discussed above.

Now, I can define $\mathrm{rep}(\mathcal A,\mathcal B)$ as the full subcategory of $\mathrm{Ho}(\mathcal A\text{-mod-}\mathcal B)$ whose objects are the quasi-functors. Moreover, I can define $\mathrm{rep}_{\mathcal V}(\mathcal A,\mathcal B)$ as the full $\mathcal V$-subcategory of $\mathcal A\text{-mod-}\mathcal B$ whose objects are the quasi-functors which are also cofibrant as bimodules.

Where can we go from here? Well, the category $\mathcal V$-$\mathbf{Cat}$ of categories enriched over $\mathcal V$ has itself a model structure: it is a known result by G. Tabuada about dg-categories, more recently generalized. The homotopy category $\mathrm{Ho}(\mathcal V\text{-}\mathbf{Cat})$ is monoidal, and $\mathrm{rep}_{\mathcal V}(\mathcal A,\mathcal B)$ gives the internal hom. In the world of dg-categories, $\mathrm{rep}(\mathcal A,\mathcal B)$ is (equivalent to) $H^0(\mathrm{rep}_{\mathcal V}(\mathcal A,\mathcal B))$.

Finally, here is my question: generalize from this particular case $\mathcal V = \mathbf C(k)$ to a more general setting, possibly letting $\mathcal V$ be a monoidal model category (with some assumptions). Every definition given above should work without problems. Has someone developed a "general theory" of those quasi-functors? I've studied it in the case of dg-categories, but I guess it is just a particular case.