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](https://arxiv.org/pdf/math/0610194.pdf), 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](https://ncatlab.org/nlab/show/Quillen+bifunctor#CoendsOverTensors) (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.



Deriving the functor $ \int_{\Gamma} F(-,-)$
We eventually found a solution which solved the original issue, though not in the way it is exactly stated here. Namely, we found what would appear to be the most general way that ends can be derived. Specifically,



> 
> **Theorem:** Let $\mathscr{C}$ be a model category and $\Gamma$ a category. Regard the functor category
>  $\mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma}$ as a model category
>  in any of the following ways:
> 
> 
> 1. as $\mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma}
>  =(\mathscr{C}^{\Gamma}\_{\mathrm{Proj}})^{\Gamma^{\mathrm{op}}}\_{\mathrm{Inj}}$
>  (assuming this model structure exists);
> 2. as $\mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma}
>  =(\mathscr{C}^{\Gamma^{\mathrm{op}}}\_{\mathrm{Proj}})^{\Gamma}\_{\mathrm{Inj}}$
>  (assuming this model structure exists);
> 3. As $\mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma}
>  =\mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma}\_{\mathrm{Reedy}}$ (assuming $\Gamma$ is a Reedy category).
> 
> 
> Then the end functor $\int\_{\Gamma}\colon
>  \mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma}\to \mathscr{C}$ is
>  right Qullen.
> 
> 
> 


Despite its simple proof, we were surprised not to find this result written anywhere, and several experts on homotopy theory claimed they had not seen it anywhere either. We therefore wrote a short paper on it, see <https://arxiv.org/abs/1807.03266>. It is very close to known results, though, and we are more than ready to accept that this result is already known and written somewhere and that we simply could not find it. If this is the case, we shall be happy to receive a reference.


