Let $f:X\to C$ be a surjective homomorphism of a regular projective surface onto a regular projective curve over a field $k$. Assume that $f(x)=c$ and that $Y\subset X$ is an integral curve such that $Y\subset f^{-1}(c)$. Then we have the following local injective homomorphisms:
- $f^{\#}_x:\mathcal O_{C,c}\to \mathcal O_{X,x}$.
- If $y$ is the generic point of $Y$, then $f^{\#}_y:\mathcal O_{C,c}\to \mathcal O_{X,y}$
- If $\mathfrak p$ is the local equation of $y$ at $x$ then $\mathcal O_{X,y}=(\mathcal O_{X,x})_{\mathfrak p}$. In particular there is an embedding $i:\mathcal O_{X,x}\hookrightarrow O_{X,y} $. See this question.
I believe that the following equality holds (just an intuition): $$f^{\#}_y=i\circ f^{\#}_x$$
But how can I prove it?
Let us assume that $x \in Y$, then this is a triviality. We have a map
$f^{\sharp}\colon \mathcal{O}_{C} \rightarrow f_{\ast}\mathcal{O}_{X}$ of sheaves on $C$. Further, we have
$\mathcal{O}_{C,c} = \lim_{\{U \subset C \mid c \in U \subset C\}} \mathcal{O}_C(U)$
$\lim_{\{U \subset C \mid x \in f^{-1}(U) \subset X\}} f_{\ast}\mathcal{O}_X(U) \rightarrow \mathcal{O}_{X,x} = \lim_{\{V \subset X \mid x \in V \subset X \}}\mathcal{O}_X(V)$
$\lim_{\{U \subset C \mid y \in f^{-1}(U) \subset X\}} f_{\ast}\mathcal{O}_X(U) \rightarrow \mathcal{O}_{X,y}= \lim_{\{V \subset X \mid y \in V \subset X\}}\mathcal{O}_X(V)$
Now the canonical maps arrise, because $c \in U \Rightarrow x \in f^{-1}(U) \Rightarrow y \in f^{-1}(U)$, and the natural maps arising from the limits above are exactly the maps you asked for.
On the other hand, if $x \notin Y$, I do not believe that what you asked is true, as the morphism $\mathcal{O}_{X,x} \rightarrow \mathcal{O}_{X,y}$ generally does not exist.