This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains finite products, kernels and cokernels; and has the property that every morphism with $0$ cokernel is an epimorphism. We call a morphism $f : A\to B$ strong if the natural map $\operatorname{coker}(\ker(A\to B)) =: \operatorname{coim}(f)\to \operatorname{im}(f) := \ker(\operatorname{coker}(A\to B))$ is an isomorphism, and we call an $\mathbb{F}_1$-linear category strong if every morphism is strong. We may consider the subcategory $\mathcal{B}^{Str}$ of $\mathcal{B}$ that has the same objects as $\mathcal{B}$, but the only morphisms in $\mathcal{B}^{Str}$ are the strong morphisms. $\mathcal{B}^{Str}$ is again an $\mathbb{F}_1$-linear category, but this time a strong one.

The motivation: In Freyd's proof of the Freyd-Mitchell embedding theorem for abelian categories, he considers the categories $$ \mathscr{L}ex\subseteq\mathscr{M}\subseteq\mathscr{F}un, $$ where $\mathcal{A}$ is an abelian category, $\mathsf{Ab}$ is the category of abelian groups, $\mathscr{L}ex$ is the category of left exact functors $F : \mathcal{A}\to\mathsf{Ab}$, $\mathscr{F}un$ is the full category of functors $F : \mathcal{A}\to\mathsf{Ab}$, and $\mathscr{M}$ is the Serre subcategory of monomorphism preserving functors. Further, a functor $T\in\mathscr{F}un$ is called torsion if for all $M\in\mathscr{M}$, $\mathrm{Hom}_{\mathscr{F}un}(T,M) = \{0\}$. Then we have the following categorization of epimorphisms in $\mathscr{L}ex$:

$L_1\to L_2\in\mathscr{L}ex$ is an epimorphism (in $\mathscr{L}ex$) if and only if the $\mathscr{F}un$-cokernel of $L_1\to L_2$ is torsion.

The question: Fix an $\mathbb{F}_1$-linear category $\mathcal{B}$, let $\mathsf{Set}_*$ denote the category of pointed sets, and let $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$ denote the category of left exact functors $F : \mathcal{B}^{Str}\to\mathsf{Set}_*$ (functors preserving finite limits).

Does there exist a subcategory $\mathscr{M}$, $$\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)\subseteq\mathscr{M}\subseteq\mathscr{F}un(\mathcal{B}^{Str},\mathsf{Set}_*),$$ (perhaps the category of monomorphism preserving functors) such that

• A morphism of left exact functors $L_1\to L_2$ is epi in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$ if and only if its cokernel $C$ in $\mathscr{F}un(\mathcal{B}^{Str},\mathsf{Set}_*)$ is torsion (i.e., $\mathrm{Hom}_{\mathscr{F}un}(C,M) = \{0\}$ for all $M\in\mathscr{M}$).

• $\mathscr{M}$ is Serre but not necessarily abelian (although $\mathscr{F}un(\mathcal{B}^{Str},\mathsf{Set}_*)$ is only $\mathbb{F}_1$-linear and not abelian, we can still define the notion of exactness in the usual way: $A\to B\to C$ is exact if $\ker(B\to C) = \operatorname{im}(A\to B)$). This condition isn't necessarily required, but it would help keep the parallel with the proof of the F-M embedding theorem in the abelian case.

If not, is there any "nice" way of categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$ (ideally, using mostly properties of the morphism as a morphism in the full functor category and avoiding properties in the category of left exact functors itself)?

This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains finite products, kernels and cokernels; and has the property that every morphism with $0$ cokernel is an epimorphism. We call a morphism $f : A\to B$ strong if the natural map $\operatorname{coker}(\ker(A\to B)) =: \operatorname{coim}(f)\to \operatorname{im}(f) := \ker(\operatorname{coker}(A\to B))$ is an isomorphism, and we call an $\mathbb{F}_1$-linear category strong if every morphism is strong. We may consider the subcategory $\mathcal{B}^{Str}$ of $\mathcal{B}$ that has the same objects as $\mathcal{B}$, but the only morphisms in $\mathcal{B}^{Str}$ are the strong morphisms. $\mathcal{B}^{Str}$ is again an $\mathbb{F}_1$-linear category, but this time a strong one.

The motivation: In Freyd's proof of the Freyd-Mitchell embedding theorem for abelian categories, he considers the categories $$ \mathscr{L}ex\subseteq\mathscr{M}\subseteq\mathscr{F}un, $$ where $\mathcal{A}$ is an abelian category, $\mathsf{Ab}$ is the category of abelian groups, $\mathscr{L}ex$ is the category of left exact functors $F : \mathcal{A}\to\mathsf{Ab}$, $\mathscr{F}un$ is the full category of functors $F : \mathcal{A}\to\mathsf{Ab}$, and $\mathscr{M}$ is the Serre subcategory of monomorphism preserving functors. Further, a functor $T\in\mathscr{F}un$ is called torsion if for all $M\in\mathscr{M}$, $\mathrm{Hom}_{\mathscr{F}un}(T,M) = \{0\}$. Then we have the following categorization of epimorphisms in $\mathscr{L}ex$:

$L_1\to L_2\in\mathscr{L}ex$ is an epimorphism (in $\mathscr{L}ex$) if and only if the $\mathscr{F}un$-cokernel of $L_1\to L_2$ is torsion.

The question: Fix an $\mathbb{F}_1$-linear category $\mathcal{B}$, let $\mathsf{Set}_*$ denote the category of pointed sets, and let $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$ denote the category of left exact functors $F : \mathcal{B}^{Str}\to\mathsf{Set}_*$ (functors preserving finite limits).

Does there exist a subcategory $\mathscr{M}$, $$\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)\subseteq\mathscr{M}\subseteq\mathscr{F}un(\mathcal{B}^{Str},\mathsf{Set}_*),$$ (perhaps the category of monomorphism preserving functors) such that

• A morphism of left exact functors $L_1\to L_2$ is epi in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$ if and only if its cokernel $C$ in $\mathscr{F}un(\mathcal{B}^{Str},\mathsf{Set}_*)$ is torsion (i.e., $\mathrm{Hom}_{\mathscr{F}un}(C,M) = \{0\}$ for all $M\in\mathscr{M}$).

• $\mathscr{M}$ is Serre but not necessarily abelian (although $\mathscr{F}un(\mathcal{B}^{Str},\mathsf{Set}_*)$ is only $\mathbb{F}_1$-linear and not abelian, we can still define the notion of exactness in the usual way: $A\to B\to C$ is exact if $\ker(B\to C) = \operatorname{im}(A\to B)$). This condition isn't necessarily required, but it would help keep the parallel with the proof of the F-M embedding theorem in the abelian case.

If not, is there any "nice" way of categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$ (ideally, using mostly properties of the morphism as a morphism in the full functor category and avoiding properties in the category of left exact functors itself)?