Lemma

Let U be an open subset of B and $s: U\rightarrow E_{\mathscr{F}}$ a continuous section of $E_{\mathscr{F}}$ over U. For every point $a$ of U, there exist an open set $V\in \mathscr{B}$ such that $a\in V$ and $V\subset U$, and an element $v$ of $\mathscr{F}(V)$ such that $s|V =\sigma_{\mathscr{F}}(V, v)$.

Let $a\in U$. By definition of the space $E_{\mathscr{F}}$, there exist an open set $V'\in \mathscr{B}$ such that $a\in V'$ and an element $t$ of $\mathscr{F}(V')$ such that $s(a) = [V', t, a]$. Then $s$ and $\sigma_{\mathscr{F}}(V', t)$ induce by restriction two continuous sections of $E_{\mathscr{F}}$ over $V'\cap U$ which are equal at the point $a$. By Prop. 11, b) of I, p. 34, there exists an open neighbourhood V of $a$, contained in $V'\cap U$, belonging to $\mathscr{B}$, such that $s$ and $\sigma_{\mathscr{F}}(V', t)$ are equal at every point of V. If we put $v=f_{VV'}(t)$, we indeed have $s|V =\sigma_{\mathscr{F}}(V, v)$.

Let us prove the proposition. For every open set U of B and every section $s\in \widetilde{\mathscr{F}}(U)$, let us denote by $D(U, s)$ the set of pairs $(V, v)$ such that $V\in \mathscr{B},V\subset U,v\in \mathscr{F}(U)$ and $s|V =\sigma_{\mathscr{F}}(V, v)$. It follows from the lemma that these open sets V form a covering of U.

If there exists a morphism $\psi :\widetilde{\mathscr{F}}\rightarrow \mathscr{G}$ such that $\psi |\mathscr{B}\circ \sigma_{\mathscr{F}}=\varphi$, then, for every open set U of B, every section $s\in \widetilde{\mathscr{F}}(U)$ and every pair $(V, v)\in D(U, s)$, we have $g_{VU}(\psi_U(s)) =\psi_V(s|V) =\varphi_V(v)$. This proves the uniqueness of $\psi$ by virtue of property $(F_1)$ of sheaves.

Let U be an open set of B and $s$ an element of $\widetilde{\mathscr{F}}(U)$. Let $(V, v)$ and $(V', v')$ be elements of $D(U, s)$. We have $s(a) = [V, v, a] = [V', v', a]$ for every point $a\in V\cap V'$. There therefore exists a pair $(W, w)\in D(V\cap V', s)$ such that $a\in W$ and $f_{WV}(v) =f_{WV}(v') =w$. We then have

and analogously,

whence

By property $(F_1)$ of sheaves, we therefore have

By properties $(F_1)$ and $(F_2)$ of sheaves, there exists a unique element $\psi_U(s)\in \mathscr{G}(U)$ such that one has:

(3) $g_{VU}(\psi_U(s)) =\varphi_V(v)$ for every $(V, v)\in D(u, s)$.

Let $\psi_U:\widetilde{\mathscr{F}}(U)\rightarrow \mathscr{G}(U)$ be the resulting mapping. It follows immediately from (3) that the family $\psi = (\psi_U)$ is a morphism of sheaves and that one has $\varphi_V=\psi_V\circ \sigma_{\mathscr{F}}(V)$ for every $V\in \mathscr{B}$.