Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega_C$ of $C$ is $\mathcal{E}xt^{n-1}_{\mathbb{P}^n}(\mathcal{O}_C,K_{\mathbb{P}^n})$ where $K_{\mathbb{P}^n}$ is the canonical sheaf on $\mathbb{P}^n$. As far as I have read (from some articles) that $\omega_C \cong \mathcal{E}xt^1_{X}(\mathcal{O}_C,K_X)$. But I do not understand why this is true. Could somebody help?