On page 3 of Beauville's book (Lemma I.5) he takes two curves $C$ and $C'$ in a surface $S$ an takes global sections $s\in H^0(S,\mathcal{O}_S(C))$ and $s'\in H^0(S,\mathcal{O}_S(C'))$. In a recent post, I was explained that you can take $s$ and $s'$ to be 1. Beauville then writes the sequence
$0\longrightarrow\mathcal{O}_S(-C-C') \xrightarrow{(s',-s)}\mathcal{O}_S(-C)\oplus \mathcal{O}_S(-C')\xrightarrow{(s,s')}\mathcal{O}_S\longrightarrow\mathcal{O}_{C\cap C'}\longrightarrow 0.$
I assume that if $\mathcal{O}_S(-C)$ (resp. $\mathcal{O}_S(-C')$) is generated on an open set $U_\alpha$ by $f_\alpha$ (resp. $f_\alpha'$), then it makes sense that the first map in the sequence takes $f_\alpha f_\alpha'$ to $(f_\alpha,-f_\alpha')$, but I don't see how this is consistent with the notation $(s',-s)$, especially since $s$ and $s'$ can be taken to be 1!
Where am I wrong here or not understanding something correctly?