I was originally unsure about this in a more basic situation: if I had a quasi-coherent ideal sheaf $\mathscr{I}$ on a scheme $X$ then on an open affine subscheme $U \subset X, U = Spec(A)$ we have $\mathscr{I} |_U = \tilde{\mathfrak{a}}$, an ideal sheaf associated to some ideal $\mathfrak{a} \subset A$. Is it true that $(\mathscr{O}_X/\mathscr{I})(U) = \mathscr{O}_X(U)/\mathscr{I}(U) = A/\mathfrak{a}$? I have a feeling that since we have an exact sequence \begin{equation} 0 \rightarrow \mathscr{I}_x \rightarrow \mathscr{O}_{X,x} \rightarrow (\mathscr{O}_X/\mathscr{I})_x \rightarrow 0 \end{equation} for all $x \in X$ hence for all $x \in U$. So this implies that we have an exact sequence on restriction \begin{equation} 0\rightarrow \mathscr{I}|_U = \tilde{\mathfrak{a}} \rightarrow \mathscr{O}_X|_U = \tilde{A} \rightarrow (\mathscr{O}_X/\mathscr{I})|_U \rightarrow 0 \end{equation} where $\tilde{A}$ is the sheaf associated to $A$ viewing as a module over itself. $\mathscr{O}_X/\mathscr{I}$ is also quasi-coherent since the other two in the exact sequence are, therefore we have $(\mathscr{O}_X/\mathscr{I})|_U = \tilde{M}$, for a sheaf associated to some $A$-module $M$. But then the exact associated sequence above will implies an exact sequence of $A$-modules \begin{equation} 0 \rightarrow \mathfrak{a} \rightarrow A \rightarrow M \rightarrow 0 \end{equation} meaning $(\mathscr{O}_X/\mathscr{I})(U) = M = A/\mathfrak{a} = \mathscr{O}_X(U)/\mathscr{I}(U)$. Apparently, because $A_f$ is flat and the argument above should be the same for any quasi-coherent sheaves $\mathscr{F}, \mathscr{G}$ such that $\mathscr{F} \subset \mathscr{G}$ we should also have $(\mathscr{F}/\mathscr{G})(V) = \mathscr{F}(V)/\mathscr{G}(V)$ for any $V\subset X$ that is contained in some affine open subset $U$. Is there any wrong with my reasoning above?
And back to main question: in general when do we have $(\mathscr{F}/\mathscr{G})(U) = \mathscr{F}(U)/\mathscr{G}(U)$?