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):

- 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.)