It is well known that the category $\mathbf{HI}_{\rm Nis}^{\rm tr}(k)$ of $\mathbb{A}^1$-local Nisnevich sheaves with transfers is closed under subobjects and quotients, from the highly nontrivial fact that $a_{\rm Nis}$ preserves $\mathbb{A}^1$-local objects.
Does it hold also for the category $\mathbf{HI}_{\rm Nis}(k)$ of $\mathbb{A}^1$-local Nisnevich sheaves without transfers?
It is an abelian category thanks to Morel's connectivity theorem but it seems nontrivial the fact that every sub sheaf of an $\mathbb{A}^1$-local sheaf (without transfers) is $\mathbb{A}^1$-local.
Of course, with rational coefficients the two categories are the same, but I was thinking of a more direct argument.
