Let $A$ be a finite presented $R$-module, so we have the sequence $$ 0 \to R^n \to R^m \to A \to 0 $$
therefore $R \to A$ gives rise for morphism of finite type $f:Spec(A) \to Spec(R)$ between affine schemes $Spec(A), Spec(R)$. By construction $\mathcal{O}_A = f_*(\mathcal{O}_R)$ is a coherent $\mathcal{O}_R $ algebra.
Now consider the dual sheaf $\underline{Hom}_{SpecR}(\mathcal{O}_A, \mathcal{O}_R)$. Obviously it's a $\mathcal{O}_{SpecR}$-module. By the locally canonical $\mathcal{O}_{A}$-action via $a' \times h \mapsto (a \mapsto h(aa'))$ the $\mathcal{O}_{SpecR}$-module $\underline{Hom}_{SpecR}(\mathcal{O}_A, \mathcal{O}_R)$ becomes an $\mathcal{O}_A$-module.
I want to see how to proof that $\underline{Hom}_{SpecR}(\mathcal{O}_A, \mathcal{O}_R)$ is a quasi coherent $\mathcal{O}_A$-module.
My idea is to show that $\underline{Hom}_{SpecR}(\mathcal{O}_A, \mathcal{O}_R)= \widetilde{Hom_A(A,R)}$ holds, using fact that on affine schemes the tilde functor provides via $M \to \widetilde{M}$ a 1-to-1 correspondence between $A$-modules and quasi coherent $\mathcal{O}_A$-modules.
I suppose that it would suffice to show that for every ideal $(p)$ the $A$-module $Hom_A(A,R)$ permutates with localisation by $(p)$, so that $$Hom_A(A,R)_{(p)} = Hom_{A_{(p)}}(A_{(p)},R_{(p)})$$
but I don't see any way how to prove it and especially how to use the information that $A$ has a finite presentation as $R$-module.
Other question: Does it suffice to show this to proof $\underline{Hom}_{SpecR}(\mathcal{O}_A, \mathcal{O}_R)= \widetilde{Hom_A(A,R)}$ and equivalently the quasi coherence of $\underline{Hom}_{SpecR}(\mathcal{O}_A, \mathcal{O}_R)$ ? The matter is how to interpret $R_{(p)}$ where $(p)$ is an ideal of $A$?