Compatible local sheaves glue uniquely up to unique isomorphism

For an open set $W\subseteq X$, define $\mathcal{F}(W)$ to be the set of families $(s_i)$ with $s_i\in {\mathcal{F}}_i(W\cap U_i)$ such that on every overlap $W\cap U_i\cap U_j$ one has ${\phi }_{ij}(s_i{\mathrm{\mid }}_{W\cap U_i\cap U_j})=s_j{\mathrm{\mid }}_{W\cap U_i\cap U_j}\mathrm{.}$ Restriction is taken componentwise. This defines a presheaf on $X$.

Fix $k$. If $W\subseteq U_k$, the map $\mathcal{F}(W)\to {\mathcal{F}}_k(W)$ sending $(s_i)$ to $s_k$ is an isomorphism: given $t\in {\mathcal{F}}_k(W)$, define $s_i:={\phi }_{ki}(t{\mathrm{\mid }}_{W\cap U_i})$, and [F1] makes these sections compatible. Therefore $\mathcal{F}{\mathrm{\mid }}_{U_k}\cong {\mathcal{F}}_k$, and the overlap maps are exactly the prescribed ${\phi }_{ij}$.

Let $W={\bigcup }_{\alpha }{W}_{\alpha }$ and let $(s_i^{(\alpha )})$ be compatible sections of the presheaf from step 1.1 on the cover. For each fixed $i$, the sections $s_i^{(\alpha )}\in {\mathcal{F}}_i(W_{\alpha }\cap U_i)$ are compatible, so [L1] glues them uniquely to a section $s_i\in {\mathcal{F}}_i(W\cap U_i)$. The cocycle condition from [F1] is preserved under these gluings, hence $(s_i)$ is a glued section of $\mathcal{F}(W)$. Uniqueness is again componentwise, so the presheaf is a sheaf. If the ${\mathcal{F}}_i$ carry abelian-group, ring, or module structures and each ${\phi }_{ij}$ preserves them, then the componentwise operations on the families $(s_i)$ make the glued sheaf a sheaf of the same kind.

If $\mathcal{G}$ is another sheaf with the same local identifications, then on every open $W$ its sections are exactly the compatible families of local sections on the cover $\{W\cap U_i\}$. By [L1], the correspondence of step 1.1 is therefore forced, so there is a unique isomorphism $\mathcal{G}\cong \mathcal{F}$. This proves uniqueness up to unique isomorphism.