Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.

- Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
- Let $Sh_{Nis}(Sm_S)\subseteq Psh(Sm_S)$ be the full subcategory of presheaves which are sheaves for the Nisnevich topology. Denote $a_{Nis}: Psh(Sm_S)^\to_\leftarrow Sh_{Nis}(Sm_S): i_{Nis}$ the inclusion / sheafification adjunction. Note that $a_{Nis}$ and $i_{Nis}$ are left exact and accessible.
- Let $Sh_{\mathbb A^1}(Sm_S) \subseteq Psh(Sm_S)$ be the full subcategory of presheaves which are $\mathbb A^1$-local. Denote by $a_{\mathbb A^1}: Psh(Sm_S)^\to_\leftarrow Sh_{\mathbb A^1}(Sm_S): i_{\mathbb A^1}$ the inclusion / localization adjunction. Again, $a_{\mathbb A^1}$ and $i_{\mathbb A^1}a_{\mathbb A^1}$, are left exact (
**EDIT**: This is the problem -- $a_{\mathbb A^!}$ is not left exact!) and accessible. - Let $Spaces_S \subseteq Psh(Sm_S)$ be the intersection $Sh_{Nis}(Sm_S) \cap Sh_{\mathbb A^1}(Sm_S)$. I'll call this
*the unstable motivic category*.

It's a fact that $Spaces_S$ is not an $\infty$-topos (cf. section 4.3 here). Nevertheless, here is a proof that it is an $\infty$-topos:

We exhibit $Spaces_S$ as a left exact localization of $Psh(Sm_S)$. Let $F: Psh(Sm_S) \to Psh(Sm_S)$ be the composite $i_{\mathbb A^1}a_{\mathbb A^1}i_{Nis} a_{Nis}$. Then $F$ is left exact and accessible. Then for any $Y \in Spaces_S$, we have that $Y$ is local with respect to the map $X \to F(X)$ for any $X \in Psh(Sm_S)$. By induction, $Y$ is also local with respect to $X \to F^\kappa(X)$ for any ordinal $\kappa$. Since $F$ is accessible, the chain $F^\bullet(X)$ eventually stabilizes at some $\kappa$; at this point we have an object of $Spaces_S$, so that the localization functor $L_{Mot}: Psh(Sm_S) \to Spaces_S$ is given by $X \mapsto F^{\infty}(X)$, where $\infty$ is the size of the universe.

Now, $L_{Mot} = F^\infty$ is a filtered colimit of left exact functors on a presheaf category, and therefore left exact. So $Spaces_S$ is an $\infty$-topos.

**Question:**
What went wrong? Is the argument flawed, or one of the premises? I've cut a few corners, but not in ways that I *think* are inherently dodgy. I'd be happy to expand on such points.