Usually, this is used in many places, so I just want to understand it properly. For a non-hyperelliptic Riemann surface $S,$ we have the canonical embedding $\iota_K:S\rightarrow \mathbb{P}^{g-1}.$ Now this is a well-known statement that a linear form in $\mathbb{P}^{g-1}$ gives a holomorphic $1$-form on $S$ and all one forms are obtained in this way. This is intuitively clear considering the form of the canonical map being $p\mapsto [\omega_1(p),\dots,\omega_g(p)]$. Now my guess is then the pull back of $\mathcal{O}(1)$ should be the canonical bundle $K$ over $S$. But I can't write a rigoruous proof. If someone explains this discussion in details that would be great! Thanks.
Edit: As one sees the transition functions of the pull back bundle should be $g_{ij}=\frac{z_j\circ \iota_K}{z_i\circ\iota_K}=\frac{\omega_j(z)}{\omega_i(z)}=g_j\frac{dz_j}{dz_i}g_i^{-1}$. So it should be isomorphic to $K$, is this correct?