I'm trying to see that if $\mathcal{F}, \mathcal{G}$ are coherent sheaves of $\mathcal R$-modules over some topological space and $\phi: \mathcal{F} \rightarrow \mathcal{G}$ is a sheaf morphism, then $\ker \phi$ is a coherent sheaf. I'm working with the definition that a sheaf is coherent if it is locally finitely generated and any sheaf of relations obtained from it is also locally finitely generated. My attempt goes as follows:
$\ker \phi$ is already a sheaf. Let $x\in X$; we find an open neighborhood $x \in U$ on which we have sections $F_1, \ldots, F_q \in \mathcal F(U)$ locally generating $\mathcal F$. This means that for $y \in U$ we can write $f_y \in F_y$ as $\sum_i g_{i,y} F_{i,y}$ with $g_{i,y} \in \mathcal R_y$ and since $\phi_y$ is $\mathcal R_y$-linear and $\phi$ commutes with taking stalks, we have that: \begin{align*} f_y \in (\ker \phi)_y &\Leftrightarrow \sum_i g_{i,y} \phi_y(F_{i,y}) = 0 \\ \Leftrightarrow \sum_i g_{i,y} (\phi(F_i))_y = 0 &\Leftrightarrow (g_{1,y}, \ldots, g_{q,y}) \in \mathcal R(\phi(F_1), \ldots, \phi(F_q))_y \end{align*} Since $\mathcal G$ is coherent, $\mathcal R( \phi(F_1), \ldots, \phi(F_q))$ is locally finite, so by shrinking $U$ if necessary we know that there exist $G_1, \ldots, G_n$ generating $\mathcal R(\phi(F_1), \ldots, \phi(F_q))_y$ at every $y \in U$. Then we can write a $(g_{1,y}, \ldots, g_{q,y}) \in \mathcal R (\phi(F_1), \ldots, \phi(F_q))_y$ as $\sum_{i=1}^n h_{i,y} G_{i,y}$.
I don't see how to proceed from here; how do I get the sections of $\ker \phi$ which locally generate it?