Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and injective model structures. Suppose that furthermore $F\colon C^{\mathrm{op}}\times D\to E $ is any bifunctor having a (point-set) right derived functor (in the sense of e.g. Shulman, definition 2.5) of the form $H(Q(-), R(-))$ for some $H\colon C^{\mathrm{op}}\times D\to E$ and cofibrant resp. fibrant replacement functors $Q, R$.

Now consider the end construction $\int_{\Gamma} F(-,-)\colon (C^{\Gamma})^{\mathrm{op}}\times D^{\Gamma} \to E$, and endow $C^{\Gamma}$ and $D^{\Gamma}$ with the projective model structures.

**Question:** Will $\int_{\Gamma} H(Q_{\mathrm{proj}}(-),R_{\mathrm{proj}}(-))\colon (C^{\Gamma})^{\mathrm{op}}\times D^{\Gamma} \to E $ be a right derived functor of $\int_{\Gamma} F$ (where $Q_{\mathrm{proj}}$ and $R_{\mathrm{proj}}$ are cofibrant resp. fibrant replacements in the projective model structure)?

In the case where $F$ is part of a two-variable Quillen adjunction, this already appears to be the case, c.f. nLab – Quillen bifunctor (one has to dualize). I am interested in whether it still holds without this assumption. At the end of the day, I am interested in the case $ F = \mathbf{dgFun}\colon\mathbf{dgCat}^{\mathrm{op}}\times\mathbf{dgCat}\to\mathbf{dgCat}$ and $ H = \mathbf{dgFun_{\infty}}$, where $\mathbf{dgFun_{\infty}}(A,B)$ has objects dg-functors $ A \to B $ and morphisms $A_{\infty}$-natural transformations between such. Here we regard $\mathbf{dgCat}$ as a 1-category endowed with Tabuada’s model structure.

**Note:** As the above question probably indicates, I am no expert in this field, I am learning it in fact, so please feel free to point out anything that might be wrong with my question.

**EDIT:** The answer in the case $F = H = \mathbf{dgFun_{\infty}}$ would also be sufficient.