Following [Anton Deitmar](http://arxiv.org/abs/1105.5290), let $\mathcal B$ be an "**$\mathbb F\_1$-linear category**" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is [balanced](http://ncatlab.org/nlab/show/balanced+category), [pointed](http://ncatlab.org/nlab/show/pointed+category), 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):


1. Are epimorphisms in $Lex(\mathcal B,\mathsf{Set}\_\*)$ also epimorphisms in $Fun(\mathcal B,\mathsf{Set}\_\*)$?
2. 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?
3. 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.)



Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?
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. 


