The residue pairing is well defined on cohomology

The functional is k-linear. Fix the global section $s$. By [F1] each summand $c\mapsto {\mathrm{res}}_p(c_{p}s)$ is $k$-linear in the principal part $c_p$ and the sum is finite for every finite-support family, so $c\mapsto {\sum }_p{\mathrm{res}}_p(c_{p}s)$ is a $k$-linear functional on the space of finite-support families of local principal parts.

Regular families are killed. Let $c$ be a family with $c_p\in {\mathcal{L}}_p$ for every closed point $p$. Since $s$ is a global section it is regular at every $p$, so $c_{p}s$ is a regular differential at $p$ by [F3] and ${\mathrm{res}}_p(c_{p}s)=0$; hence the functional vanishes on $c$.

Principal parts of meromorphic sections are killed. Let $\theta$ be a global meromorphic section of $\mathcal{L}$, that is, an element of ${\mathcal{L}}_{\eta }$, and let $c=(c_p)$ be its family of local principal parts $c_p=\theta +{\mathcal{L}}_p$. By [F5] the product $\theta s$ is a rational differential on $C$, and its local principal part at $p$ is $c_{p}s$, so ${\sum }_p{\mathrm{res}}_p(c_{p}s)={\sum }_p{\mathrm{res}}_p(\theta s)$. By the global residue theorem [F4] the right-hand sum vanishes, so the functional kills the principal parts of every global meromorphic section of $\mathcal{L}$.

Descend to cohomology. By [F2] the kernel of the surjection from finite-support families of local principal parts onto $H^1(C,\mathcal{L})$ is generated by the two kinds of families treated in steps 1.2 and 1.3, so the $k$-linear functional of step 1.1 factors through a $k$-linear functional $H^1(C,\mathcal{L})\to k$. Letting $s$ vary, the construction is $k$-linear in $s$ by [F1] and natural in $\mathcal{L}$ because the product and the residue are, so the pairing $⟨,⟩\text{\hspace{0.17em}⁣}:H^1(C,\mathcal{L})\times H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})\to k$ is well defined and $k$-bilinear; this is the independence statement of the definition of the pairing, and it is exactly the global residue theorem applied to $\theta s$.