Let $C$ be an Gorenstein(so the dualizing sheaf $\omega_{C}$ is invertible),
integral curve (therefore $1$-dimensional, proper $k$-scheme over a fixed field), $\mathcal{L} \in Pic(C)$ ample,
$f: C \to \mathbb{P}^n _k$ a proper morphism induced via $\mathcal{L}:=f^*(\mathcal{O}_{\mathbb{P}^n}(1))$ and the image
$C' := f(C) = im(f) \subset \mathbb{P}^n$ an integral curve of $\mathbb{P}^n$.
We can factorise $f$ als $C \xrightarrow{f'} C' \xrightarrow{\iota} \mathbb{P}^n$. By definition we have $\mathcal{O}_{C'}(1) := \iota^* \mathcal{O}_{\mathbb{P}^n}(1)$.
My question is why $H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(1)) \cong H^0(C', \mathcal{O}_{C'}(1)) \cong H^0(C, \mathcal{L})$ holds?