I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme. 


Let $E$ be an $\Omega$-spectrum and $\varphi \colon F\to E$ be a morphism of spectra satisfying that for any smooth $S$-scheme $X$ the induced map
$$
\mathrm{Hom}\_{\mathbf{SH}(S)}(X,F)\longrightarrow \mathrm{Hom}\_{\mathbf{SH}(S)}(X,E)
$$ 
is an isomorphism of groups. The question is: **Is $\varphi\colon F\to E$ an isomorphism of spectra?**



Basic questions on spectra
Here is a slightly more fleshed out version of the comment above. 


First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of compact generators, is Theorem 9.1 of <https://arxiv.org/pdf/math/0310190.pdf> (there may be earlier references). In particular, the smallest localizing subcategory of $SH(S)$ generated by $\mathcal{C}$ is $SH(S)$ itself. 


Now let $\phi \colon F \to E$ be as in your statement, with cofiber $Z$, and assume that the induced map $Hom\_{SH(S)}(X,F) \to Hom\_{SH(S)}(X,E)$ is a $\pi\_{\*,\*}$ equivalence. It follows that $[X,Z] = 0$ for all $X \in \mathcal{C}$. Since $\mathcal{C}$ generates, this implies that $Z \simeq \ast$. 


