Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category? Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and cokernels, and has the property that every morphism with zero cokernel is an epimorphism. Let $\mathsf{Set}_*$ denote the category of pointed sets, and let $Lex(\mathcal B,\mathsf{Set}_*)$ denote the category of left-exact functors $F: \mathcal B\to\mathsf{Set}_*$ (that is, $F$ preserves finite limits).
I want to show that $Lex(\mathcal B,\mathsf{Set}_*)$ is also $\mathbb F_1$-linear if $\mathcal B$ is small; in particular, the properties that still concern me are that $Lex(\mathcal B,\mathsf{Set}_*)$ is balanced, and that every morphism with $0$ cokernel is an epimorphism. I know that $Lex(\mathcal B,\mathsf{Set}_*)$ is a reflective subcategory of $Fun(\mathcal B,\mathsf{Set}_*)$ and that finite products, kernels, and cokernels exist in $Lex(\mathcal B,\mathsf{Set}_*)$ (it's also clear that $Lex(\mathcal B,\mathsf{Set}_*)$ is pointed), and I know that $Fun(\mathcal B,\mathsf{Set}_*)$ is $\mathbb F_1$-linear. My hope was originally to prove that if a natural transformation $\eta : F\to G$ is mono/epi in $Lex(\mathcal B,\mathsf{Set}_*)$, then it is mono/epi in the full functor category, which would show that $Lex(\mathcal B,\mathsf{Set}_*)$ is balanced (and I had a similar plan to show that morphisms with $0$ cokernel were epi), although a proof of this is turning out to be quite elusive (and I'm no longer convinced that one exists). I was able to show that if a natural transformation is a monomorphism in $Lex(\mathcal B,\mathsf{Set}_*)$, it is a monomorphism in the full functor category - epimorphisms are the root of my problems. Then my question is really a few related questions (hopefully they're related enough to warrant posting all of them here):


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*Are epimorphisms in $Lex(\mathcal B,\mathsf{Set}_*)$ also epimorphisms in $Fun(\mathcal B,\mathsf{Set}_*)$?

*If not, does anyone know of an alternate proof (or a resource where I could find one) that a bimorphism in $Lex(\mathcal B,\mathsf{Set}_*)$ is also an isomorphism?

*Is there any reasonable condition one can place on $\mathcal B$ to force epis in $Lex(\mathcal B,\mathsf{Set}_*)$ to remain epis in $Fun(\mathcal B,\mathsf{Set}_*)$ or to force bimorphisms in $Lex(\mathcal B,\mathsf{Set}_*)$ to be isomorphisms? (I doubt that this has an answer in the affirmative, as my experience indicates that many of the properties of a functor category depend on the codomain rather than the domain.)

 A: Here is a counterexample (modulo a small statement I'm not sure how to prove). Building on the comments, let $B$ be (a skeleton of) the opposite of the category of at most countable groups. This satisfies all of the desired conditions. There is a functor $\text{Grp} \to \text{Lex}(B, \text{Set}_{\ast})$ sending a group $G$ to the representable functor $\text{Hom}(-, G)$ (which lands in $\text{Set}_{\ast}$ since $\text{Grp}$ has a zero object) which is full and faithful; you can see this by evaluating on the subcategory given by the free groups, which gives a right adjoint $\text{Lex}(B, \text{Set}_{\ast}) \to \text{Grp}$ (this is the statement I'm not sure how to prove). In particular, $\text{Grp}$ is a coreflective subcategory, so colimits in $\text{Grp}$ agree with those in $\text{Lex}(B, \text{Set}_{\ast})$. 
But $\text{Grp}$ has morphisms with zero cokernel which are not epimorphisms: take any inclusion $f : H \to G$ of a proper subgroup $H$ into a group $G$ whose normal closure is all of $G$ (for example, take $G$ to be simple and $H$ to be proper and nontrivial). Hence the same is true of $\text{Lex}(B, \text{Set}_{\ast})$. 
For the purposes of imitating the proof of Freyd-Mitchell I think you should be looking at $\text{Lex}(B, \text{Set}_{\ast})^{op}$ as I mentioned in the comments. 
