The context for this question comes from [this arxiv preprint](https://arxiv.org/abs/2210.07753). Specifically, a remark in the final proof of the paper. To make the question more self-contained, I'll phrase this question in a slightly more general setting.


Fix a category $A$ and a combinatorial model category $\mathcal{M}$ where all objects are cofibrant [1]. Given a pair objects $X : \mathcal{M}$ and $a : A$, we define $a \otimes X$ to be the left Kan extension of $X$ along the inclusion $\{a\} \to A$. Explicitly then,


$$
 (a \otimes X)(a') = \hom(a,a') \cdot X
$$


**The question:** Endowing $\mathbf{Fun}(A,\mathcal{M})$ with the projective model structure, can an arbitrary functor $F : A \to \mathcal{M}$ be built out of an (iterated) homotopy colimit of $a \otimes X$s?


As a first step, we can use the classical Yoneda formula to realize $F$ as $\int^{a : A} a \otimes F(a)$, so $F$ can certainly built as the coequalizer of coproducts of presheaves of the form $a \otimes X$. I'm unsure how to argue that this is the homotopy colimit of some functor. It seems natural enough to extend this coequalizer formula to a simplicial object:


$$
 G(n) = \coprod\_{a\_0 \to \dots \to a\_n} a\_0 \otimes F(a\_n)
$$


This then appears to be a cofibrant diagram with respect to the Reedy model structure on simplicial objects of $\mathbf{Fun}(A, \mathcal{M})$ whose colimit is $F$. However, my understanding is that this alone does *not* imply that $\mathsf{hocolim}\,G = F$ since $\Delta : \mathbf{Fun}(A, \mathcal{M}) \to \mathbf{Fun}(\Delta^\mathsf{op}, \mathbf{Fun}(A, \mathcal{M}))$ is not necessarily a right Quillen functor.


Is it still the case that $\mathsf{hocolim}\,G = F$? Or do I need to search for a different $G$ entirely? I asked a similar question on a different forum and was adviced to use the Bousfield-Kan formula. This certainly appears to be quite related but I'm not quite sure how to join the pieces together.


[1]: For the actual example, $\mathcal{M}$ is $\mathbf{sSet}^\sharp\_{/B}$ with the Cartesian model structure.



Decomposing a $\mathcal{M}$-valued presheaf into a homotopy colimit of representables
The diagram $G$ is a projectively cofibrant diagram because
nondegenerate simplices split off as a coproduct summand in every degree
and every simplicial level is a coproduct of (enriched) representables.


The homotopy colimit of a projectively cofibrant diagram can be computed as its colimit.
Therefore, the homotopy colimit of $G$ can be computed as $F$.


