Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote the $\infty$-category of small stable $\infty$-categories and exact functors by $Cat^{\rm ex}\_{\infty}$. Suppose I have given a functor 
$$
F \colon \mathcal{I} \times \mathcal{I}^{\rm op} \to Cat^{\rm ex}\_{\infty}
$$
where I am secretly regarding the category on the left as an $\infty$-category.



> 
> Are coends defined for $\infty$-functors as above? Does the coend of $F$ as above always exist? 
> 
> 
> If the answer to the above questions is "yes": Can we describe the weak Kan complex underlying the coend in some way?
> 
> 
> 



coends of stable infinity categories
There is some discussion of coends in $\infty$-categories in [my PhD thesis](http://arxiv.org/abs/1011.3243) (Sheffield, 2010). I think that, given that the target category $\mathrm{Cat}\_\infty^\mathrm{ex}$ is cocomplete, this gives you one way to do it.


I have another way, which is arguably rather less fussy, but it is currently not written up in a presentable fashion.


