Given projective curves $C$ and $C'$ and a surjective morphism $\varphi\colon C\to C'$, such that $Q\in C'$ is a smooth point and its fibre $\varphi^{-1}(Q)$ consists of smooth points.

Then $\mathcal{O}_{C',Q}$ and $\mathcal{O}_{C,P}$, for $\varphi(P) = Q$, are discrete valuation rings.

My question is: Are all discrete valuation rings containing $\mathcal{O}_{C',Q}$ of the form $\mathcal{O}_{C,P}$, for some $P\in\varphi^{-1}(Q)$?

If this is not true under this conditions, can one make additional assumptions, such that this is true? And a counterexample, if my claim is wrong, would be nice.

I already asked this on stackexchange, but didn't get any answer.