Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $d$ with a canonical module $\omega.$ Let $M,N$ be maximal Cohen-Macaulay $R$-modules. Then local duality implies $$\mathrm{Ext}^i(N,\mathrm{Hom}(M,\omega))=\mathrm{Ext}^i(M,\mathrm{Hom}(N,\omega)).$$
It will be really helpful if someone explains the above equality using local duality.
(Here by local duality I mean $\mathrm{Hom}(\mathrm{Ext}^i(M,\omega),E)\cong H_{\mathfrak m}^{d-i}(M)$ where $E$ is the injective hull of residue field of $R.$)