In a $k$-linear abelian category $\mathscr{A}$, where $k$ is a field, two objects $A,B$ are given. The extension group $\text{Ext}^1_{\mathscr{A}}(A,B)$ is a group consisting of the exact sequences $$0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0,$$
where $E$ is an object in $\mathscr{A}$, together with the Baer sum as a binary operation. To my knowledge, the Baer sum works as follows. Let $$0\rightarrow B\overset{f_1}{\rightarrow} E_1\overset{g_1}{\rightarrow} A\rightarrow 0,$$ and
$$0\rightarrow B\overset{f_2}{\rightarrow} E_2\overset{g_2}{\rightarrow} A\rightarrow 0.$$
Denote th sum of the two exact sequences
$$0\rightarrow B\overset{f_3}{\rightarrow} E_3\overset{g_3}{\rightarrow} A\rightarrow 0,$$
where the middle object is defined as {$(e_1,e_2)\in E_1\oplus E_2 | g_1(e_1)=g_2(e_2)$} by quotienting by {$(f_1(b),-f_2(b))\in E_1\oplus E_2|b\in B$} and where $f_3$ maps $b\in B$ to $(f_1(b),0)\sim (0,f_2(b))$ and $g_3$ maps $(e_1,e_2)\in E_3$ to $g_1(e_1)=g_2(e_2)$.
$k$-linearity of an abelian category means, that every morphism set between objects is a $k$-module, i.e. and the composition maps are $k$-bilinear.
Consider now the functor $Hom_{k}(-,k)$ mapping from some abelian category $\mathscr{B}$ to $k$, where objects in $\mathscr{B}$ is mapped to $Hom_{k}(A,k)$ and for any objects $C,D\in\text{obj}(\mathscr{B})$ the morphisms $\phi:C\rightarrow D$, which are $k$-linear, map under this contravariant functor a $k$-module $Hom_{k}(D,k)$ to $\text{Hom}_{k}(C,k)$ with the rule $\alpha\mapsto \alpha\circ\phi$ by reversing the arrow. Being more specific, the extension group is $k$-linear and forms therefore an $k$-module or rather a $k$-vector space and live in the category $\text{Vect}_k$, which is also an abelian category.
I am studying currently Serre-duality, but the following question is just a first step to understand the whole statement. How do I define this morphism from the extension group to an element of $k$? Maybe it is a bit speculative, but I thought that possibly the functor takes a sequence $0\rightarrow B\overset{f}{\rightarrow} E\overset{g}{\rightarrow} A\rightarrow 0$,under a map $\psi\in\text{Hom}_{k}(\text{Ext}^1_k(A,B),k)$ to $\psi_1\circ f+\psi_2\circ g$ with $\psi=(\psi_1,\psi_2)$ and $\psi_1\in\text{Hom}(E,k)$ and $\psi_2\in\text{Hom}(A,k)$. Is such a morphism compatible with the Baer sum? A difficulty might also be to define this morphism in a well-defined way.