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The following excerpt is from an english translation of Serre's FAC (here). This is in Chapter I: Sheaves, $\S1:$ Operations on Sheaves, n$^{\circ}$ 7: Subsheaf and Quotient Sheaf and n$^{\circ}$ 8: Homomorphisms, Page 13.

$\textbf{Subsheaf and Quotient Sheaf:}$

....

The second definition of $\mathscr{F}$/$\mathscr{G}$ shows that if $s$ is a section of $\mathscr{F}$/$\mathscr{G}$ over a neighborhood of $x$,

there exists $t$ of $\mathscr{F}$ over a neighborhood of $x$ such that the class of $t(y)$ $\mathrm{mod}$ $\mathscr{G}_y$ is equal to $s(y)$ for all $y$ close enough to $x$.

...

$\textbf{Homomorphisms:}$

Let $\mathscr{A}$ be a sheaf of rings, $\mathscr{F}$ and $\mathscr{G}$ two sheaves of $\mathscr{A}-$modules. An $\mathscr{A}-$homomorphism from $\mathscr{F}$ to $\mathscr{G}$ is given by, for all $x \in X$, an $\mathscr{A}_x-$ homomorphism $\phi_x \colon \mathscr{F}_x \to \mathscr{G}_x$,

such that the mapping $\phi \colon \mathscr{F} \to \mathscr{G}$ defined by the $\phi_x$ is continuous. This condition can also be expressed by saying that, if $s$ is a section of $\mathscr{F}$ over $U$, the $\phi \circ s$ is a section of $\mathscr{G}$ over $U$.

For example, if $\mathscr{G}$ is a subsheaf of $\mathscr{F}$, the injection $\mathscr{G} \to \mathscr{F}$ and the projection $\mathscr{F} \to \mathscr{F}$/$\mathscr{G}$ both are homomorphisms.

$\textbf{Proposition 7.}$ Let $\phi$ be a homomorphism from $\mathscr{F} \to \mathscr{G}$. For all $x \in X$, let $\mathscr{N}_x$ be the kernel of $\phi_x$ and let $\mathscr{I}_x$ be the image of $\phi_x$. Then $\mathscr{N} = \bigcup \mathscr{N}_x$ is a subsheaf of $\mathscr{F}$, $\mathscr{I} = \bigcup \mathscr{I}_x$ is a subsheaf of $\mathscr{G}$,

and $\phi$ defines an isomorphism of $\mathscr{F}$/$\mathscr{N}$ and $\mathscr{I}$.

Proof. Since $\phi_x$ is an $\mathscr{A}_x-$homomorphism, $\mathscr{N}_x$ and $\mathscr{I}_x$ are submodules of $\mathscr{F}_x$ and $\mathscr{G}_x$ respectively, and $\phi_x$ defines an isomorphism of $\mathscr{F}_x$/$\mathscr{N}_x$ with $\mathscr{I}_x$. If on the other hand $s$ is a local section of $\mathscr{F}$ such that $s(x) \in \mathscr{N}_x$, we have $\phi \circ s(x) = 0$, hence $\phi \circ s(y) = 0$ for $y$ close enough to $x$, so $s(y) \in \mathscr{N}_y$ thus $\mathscr{N}$ is a subsheaf of $\mathscr{F}$. If $t$ is a local section of $\mathscr{G}$, such that $t(x) \in \mathscr{I}_x$, there exists a local section $s$ of $\mathscr{F}$, such that $\phi \circ s(x) = t(x)$, hence $\phi \circ s = t$ in a neighborhood of $x$, showing that $\mathscr{I}$ is a subsheaf of $\mathscr{G}$, isomorphic with $\mathscr{F}$/$\mathscr{N}$.

Question: I am unsure where or how the continuity of $\phi$ and $\phi^{-1}$ between $\mathscr{F}$/$\mathscr{N}$ and $\mathscr{I}$ is determined. I understand every part of Serre's argument, except he doesn't mention why the big map is continuous? I of course see why we have the isomorphisms $\phi_x$ he mentions, but why is $\phi$ continuous? and $\phi^{-1}$? Is there an obvious reason why the first highlighted criterion holds? Incase it is unclear, the other stuff he is doing in the proof is checking that the two objects are subsheaves using a definition and characterization he gives on the previous page (page 11). All the stuff I highlighted above is the stuff that is related to my question, or stuff I thought might help with the solution.

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  • $\begingroup$ I think I got it.. send s(U) up into the image sheaf then pull back into big F along phi and project into the quotient, the resulting set is open and should be equal to the preimage of s(U) in the quotient $\endgroup$ – Prince M Oct 29 '18 at 23:33

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