Let $\text{Cart}$ be the category of cartesian spaces which has as its objects the collection of sets $U$ for which there exists $n \in \mathbb{N}$ so that $U \subset \mathbb{R}^n$ and $U$ is diffeomorphic to $\mathbb{R}^n$. The category $\text{Cart}$ turns into a site by endowing it with the coverage of good open covers, i.e., open covers $\mathcal{U} = \{ U_j \}_{j \in \mathcal{J}}$ such that $U_j \cap U_k$ is either empty or diffeomorphic to $\mathbb{R}^n$. For any good cover $\mathcal{U}$ of an object $V \in \text{Cart}$ as above we may then define the Cech nerve $c^\mathcal{U}$ as the simplicial presheaf on $\text{Cart}$ with $m$-simplices $$c^\mathcal{U}_m = \coprod\limits_{\zeta \colon m} j_\text{Cart}(U_\zeta)$$ where, first of all, $\zeta \colon m$ should mean that $\zeta$ runs over all those $(m+1)$-tuples $(\zeta_0, \ldots, \zeta_m) \in \mathcal{J}^{m+1}$ for which we have $$U_\zeta \equiv \bigcap\limits_{i = 0}^m U_{\zeta_i} \neq \emptyset$$ Moreover, the functor $j_\text{Cart} \colon \text{Cart} \to \text{Psh}_\Delta(\text{Cart})$, which has as its codomain the category of simplicial presheaves on $\text{Cart}$, takes a cartesian space $U \in \text{Cart}$ and maps it onto the Yoneda embedding of the object $U$ but viewed as a constant simplicial object in the category of presheaves on $\text{Cart}$. There is then a unique map induced by the universal property of the respective coproducts: $$\psi^\mathcal{U} \colon c^\mathcal{U} \to j_\text{Cart}(V)$$ The Cech model structure on the category of presheaves is then defined to be the the left Bousfield localization of the injective model structure $\text{Psh}_\Delta(\text{Cart})_\text{inj}$ with respect to the family of morphisms $$\psi^\mathcal{U} \colon c^\mathcal{U} \to j_\text{Cart}(V)$$ Call this new model structure $\text{Psh}_\Delta(\text{Cart})_\text{loc}$. Fibrant objects are precisely those objects $X \in \text{Psh}_\Delta(\text{Cart})$ which are fibrant with respect to the injective model structure and for which $$\mathbb{R}\text{Map}(\psi^\mathcal{U}, X) \colon \mathbb{R}\text{Map}(j_\text{Cart}(V), X) \to \mathbb{R}\text{Map}(c^\mathcal{U},X)$$ is a weak equivalence of simplicial sets, where $\text{Map}$ is the simplicially enriched hom-functor and $\mathbb{R}\text{Map}$ is the right derived version of the former.
How does one deduce that the above weak equivalence boils down to the statement that we have a weak equivalence $$X(V) \overset{\sim}{\to} \text{holim}_\zeta X(U_\zeta)$$
My attempt is this: Since $X$ is fibrant in the injective model structure and any object is cofibrant with respect to the injective model structure, it suffices to consider $$\text{Map}(\psi^\mathcal{U}, X) \colon \text{Map}(j_\text{Cart}(V), X) \to \text{Map}(c^\mathcal{U},X)$$ The domain of this map certainly boils down to $X(V)$ by a simple application of the Yoneda Lemma. The right hand-side is seen to be naturally isomorphic to $$\prod\limits_{\zeta \colon -} \text{Map}(U_\zeta, X) \cong \prod_{\zeta\colon -} X(U_\zeta)$$ This looks already pretty good. It seems as if this thing should boil down to a homotopy limit, but i don't really see how to make this rigorous. Any help is welcome :)
Any simplicial presheaf is a homotopy colimit (over $\def\op{{\sf op}}Δ^\op$) of its individual layers, which are presheaves of sets.
Thus, we have $$\def\R{{\bf R}}\def\Map{\mathop{\rm Map}}\def\holim{\mathop{\rm holim}}\def\hocolim{\mathop{\rm hocolim}}\R\Map(c^U,X)≃\R\Map(\hocolim_{n∈Δ^\op}c^U_n,X)≃\holim_{n∈Δ}\Map(c^U_n,X).$$
Next, since every $c^U_n$ is a coproduct (hence homotopy coproduct) of representables, we have $$\def\ho{{\sf h}}\holim_{n∈Δ}\Map(c^U_n,X)≃\holim_{n∈Δ}\R\Map(\coprod^\ho_{ζ:n}j(U_ζ),X)≃\holim_{n∈Δ}\prod^\ho_{ζ:n}\R\Map(j(U_ζ),X)≃\holim_{n∈Δ,ζ:n}X(U_ζ),$$ as desired.