I am interested in additive categories appearing in functional analysis, in particular, the category $LCS$ of locally convex spaces and continuous linear functions. This category is not abelian but quasi-abelian, i.e., the canonical morphism between the coimage and the image is always monic and epic but not necessarily an isomorphism and, moreover, kernels push out to kernels and cokernels pull back to cokernels. More importantly, this category has enough injective objects and the construction of right derivatives of covariant additive functors with values in an abelian (or quasi-abelian) category works as in the abelian case (this has been done by Palamodov at the end of the sixties).
On the other hand, $LCS$ only has very few projective objects so that there are no left derivatives of contravariant functors. One can however consider a contravariant functor $F$ from $LCS$ to an abelian category $A$ as a covariant functor $F:LCS\to A^{op}$ which thus has right derivatives.
So far, so good. But the first example coming to mind is a contravariant hom-functor $Hom(-,T)$. Considered as a covariant functor $LCS \to Ab^{op}$ (the opposite category of abelian groups) this looks rather strange, for example it is not left-exact (which seems to be a standard assumption when introducing right derivatives but, as far as I see, the definition makes sense for any additive functor, the $0$-th derivative then does not coincide with the functor itself).
I do not expect that these weird derivatives are useful nor that the question has really much to do with locally convex spaces.
Nevertheless, I want to ask: What are the relations to the usual $Ext^n$-functors (the right derivatives of $Hom(S,-)$)?
