For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules over the motivic cobordism and motivic cohomology spectra (the latter are essentially Voevodsky motives); these categories also contain motivic spectra for all smooth varieties over $k$ that I will denote by $\Sigma^{\infty}(-_+)$. For an object $N$ of the corresponding homotopy $t$-structure, a smooth $X/k$, and $n\ge 0$ I would like to compute the morphism group $Hom(\Sigma^{\infty}(X_+),N[n])$ (in each of the aforementioned categories).

Now, to the object $N$ one can associate the presheaf $S_N$ sending a smooth $Y/k$ into $Hom(\Sigma^{\infty}(X_+),N)$; actually, $S_N$ is a Nisnevich sheaf.

I believe that $Hom(\Sigma^{\infty}(X_+),N[n])$ equals $H^n_{Nis}(X,S_N)$. Is there any reference for this fact (for any of the aforementioned motivic categories)? A natural way to prove it would be to start with the stable homotopy category $SH_{Nis}^{S^1}(k)$ of simplicial Nisnevich sheaves (of sets on smooth varieties over $k$) and to study adjoint functors connecting this category with the "motivic categories"; yet I don't have fine enough references for this argument.

For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules over the motivic cobordism and motivic cohomology spectra (the latter are essentially Voevodsky motives); these categories also contain motivic spectra for all smooth varieties over $k$ that I will denote by $\Sigma^{\infty}(-_+)$. For an object $N$ of the heart of the corresponding homotopy $t$-structure, a smooth $X/k$, and $n\ge 0$ I would like to compute the morphism group $Hom(\Sigma^{\infty}(X_+),N[n])$ (in each of the aforementioned categories).

Now, to the object $N$ of the heart one can associate the presheaf $S_N$ sending a smooth $Y/k$ into $Hom(\Sigma^{\infty}(X_+),N)$; actually, $S_N$ is a Nisnevich sheaf.

I believe that $Hom(\Sigma^{\infty}(X_+),N[n])$ equals $H^n_{Nis}(X,S_N)$. Is there any reference for this fact (for any of the aforementioned motivic categories)? A natural way to prove it would be to start with the stable homotopy category $SH_{Nis}^{S^1}(k)$ of simplicial Nisnevich sheaves (of sets on smooth varieties over $k$) and to study adjoint functors connecting this category with the "motivic categories"; yet I don't have fine enough references for this argument.