I have a concrete complete and cocomplete category $\mathcal{A}$ (in fact, a category of modules over a monoid in a symmetric closed monoidal category) in which all monomorphisms and all epimorphisms are regular. However, the initial object $i$ and the terminal object $t$ do not coincide, whence I should not be able to speak about zero morphism, kernels or cokernels.
Despite of this, I wonder if one can still speak about (short) exact sequences by using, for example, kernel pairs and set-theoretic images (btw: it happens that set-theoretic images of morphisms in $\mathcal{A}$ are in $\mathcal{A}$ again).
I tried to search in the literature but I didn't find anything in this concern. Has this question been addressed already somewhere, even in different forms?
Edit: After checking Arnaud's suggested paper, I realized that what Bourn does there cannot be exactly applied to my context (or, at least, it does not provide useful information). Let me specify that the inital object in my category $\mathcal{A}$ has as underlying set the empty set $\emptyset$, while the terminal one has the singleton $*$. By applying Bourn's construction, I would have that all the kernels are empty, while the kernel pairs are not.