It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it.


I'm sure this can be done not only for $\text{Set}$-enriched categories but for general $\mathcal{V}$-enriched categories, with the appropriate notions of $\mathcal{V}$-enriched colimit, and functor preserving the enrichment, and I just found it in section 4.4. of Kelly's Basic Concepts of Enriched Category Theory.


So now the question is: can one prove proposition 2.3 in this [paper](https://arxiv.org/pdf/math/0007070.pdf) about simplicial pre-sheaves on $\mathcal{C}$ being the universal model category on $\mathcal{C}$ by just doing the $\mathcal{V}$-enriched co-completion with $\mathcal{V}=\text{sSet}$?



Universal model category as a $\text{sSet}$-enriched co-completion
Given $C$ a small category (eventually, a small simplicial category) I denote by $UC$ the projective model structure on the category of simplicial presheaves on $C$ as in the paper. Using the kind of argument you have in mind we obtain the following theorem:


**Theorem:** If $M$ is a simplicial model category, then there is an equivalence of categories between:


* (Simplicial) Functors $C \to M$ taking values in the full subcategory of cofibrant objects.
* *Simplicial* left Quillen functor $UC \to M$.


In one direction, the equivalence is simply given by restricting to the Yoneda embedding $ C \to UC$ as representable are cofibrant in the projective model structure, this forces the composite functor $C \to UC \to M$ to take values in cofibrant objects. In the converse direction, one takes the unique simplicial left adjoint functor $UC \to M$ and check, using the axiom of simplicial model category for $M$ that this is a left Quillen functor.


However, this is *not* what the paper you mention proves.


There, they start from a model category $M$ that is not assumed to be a simplicial model category, and a functor $C \to M$ not assumed to takes values in cofibrant objects. And construct a left Quillen functor $UC \to M$ by considering (and choosing) a cofibrant simplicial resolution of the functor $C \to M$ they started from. In particular, the "uniqueness" of the left Quillen functor obtained this way, is only up to homotopy (to be more precise, up to a contractible space of choices).




---


One abstract way to understand the relation between the two is as follows:


Given $M$ a combinatorial left proper model category, there is a Quillen equivalent simplicial model structure on the category $sM$ on the category of simplicial objects of $M$, (this is explained in the [paper](https://www.ams.org/journals/tran/2001-353-12/S0002-9947-01-02661-7/S0002-9947-01-02661-7.pdf) "Replacing model categories with simplicial one" by Dugger)


The evaluation at $[0]$ gives a left Quillen equivalence $sM \to M$


One way to understand the non-simplicial theorem is that if you start from $C \to M$, you can see it as a functor $C \to sM$ taking values in constant simplicial objects, then take a levelwise cofibrant replacement to obtain a functor $C \to sM$ taking value in cofibrant object, apply the "simplicial theorem" to get a Quillen functor $UC \to sM$ and finally, post compose with Quillen functor $sM \to M$ that evaluate at $[0]$.


Now for the model structure on $sM$ to exist we need $M$ to be combinatorial and left proper, if you are willing to work with a left semi-model structure instead it is enough to assume that $M$ is an accessible model category (no properness assumption).


But in some sense the central observation of the paper you quote, is that, even if the model structure on $sM$ cannot be constructed, the overall construction make sense with no assumption $M$ (other than being a model category, I guess they also need functorial factorization, I do not remember).


