I am working on understanding the proof of the following proposition given in Serre's FAC. The following is verbatim from page 11.

Proposition 6. Let $Y$ be a closed subspace of $X$, and let $\mathscr{G}$ be a sheaf on $Y$. Put $\mathscr{F}_{x} = \mathscr{G}_{x}$ for $x \in Y$, and $\mathscr{F}_x = 0$ for $x \notin Y$, and let $\mathscr{F}$ be the sum of the sets $\mathscr{F}_x$. Then $\mathscr{F}$ admits a unique structure of a sheaf over $X$ such that $\mathscr{F}(Y)=\mathscr{G}$.

Proof (Uniqueness). Let $U$ be an open subset of $X$; if $s$ is a section of $\mathscr{G}$ on $U \cap Y$, extend $s$ by 0 on $U$-$(U \cap Y)$. When $s$ runs over $\Gamma(U \cap Y, \mathscr{G}$), we obtain in this way a group $\mathscr{F}_U$ of mappings from $U$ to $\mathscr{F}$. Proposition 5 then shows that if $\mathscr{F}$ is equipped with the structure of a sheaf such that $\mathscr{F}(Y) = \mathscr{G}$, we have $\mathscr{F}_{U} = \Gamma(U,\mathscr{F})$ which proves the uniqueness of the structure in question.

Proposition 5. If a sheaf $\mathscr{F}$ is concentrated on a closed subspace $Y$, the homomorphism $$\rho_{Y}^{X}\colon \Gamma(X,\mathscr{F}) \to \Gamma(Y,\mathscr{F}(Y))$$ is bijective.

My understanding of the assertion in proposition 6. What we are saying is that for a top space $X$, and a closed subspace $Y$, if we have some sheaf $\mathscr{G}$ on $Y$ then there is exactly one way to extend the sheaf by 0 such that restricting back to $Y$ recovers the sheaf $\mathscr{G}$. Since I am only focusing on the uniqueness, what this should mean is that if I have $\mathscr{F}_1$ and $\mathscr{F}_2$ that are constructed as the sum of the sets $\mathscr{F}_x$ as described, then if I augment $\mathscr{F}_1$ and $\mathscr{F}_2$ with sheaf structures (a topology that makes the projection map a local homeomorphism, and group operations continuous) such that restricting to $Y$ gives $\mathscr{G}$, then $\mathscr{F}_1$ and $\mathscr{F}_2$ are isomorphic, which should mean the existence of a homeomorphism between them that commutes with projection maps?

My understanding of Serre's proof. Basically nothing, in particular how is he applying proposition 5? It seems he is showing that $\Gamma(U,\mathscr{F})$ is uniquely determined, but can someone elaborate on his argument for me?

Slightly unrelated question: In the situation that $\mathscr{F}_{x} = \mathscr{G}_{x}$ for $x \in Y$, and $\mathscr{F}_x = 0$ for $x \notin Y$, and let $\mathscr{F}$ be the sum of the sets $\mathscr{F}_x$, can someone give an example of a sheaf structure on $\mathscr{F}$ so that $\mathscr{F}(Y)$ is not $\mathscr{G}$?


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