Corollary 2

Let B be a topological space, $\mathscr{F}$ and $\mathscr{G}$ sheaves on B and $\varphi$ a morphism of sheaves from $\mathscr{F}$ into $\mathscr{G}$. The following assertions are equivalent:

(i) $\varphi$ is an isomorphism;

(ii) There exists a basis $\mathscr{B}$ of the topology of B such that for every $U\in \mathscr{B}$, the mapping $\varphi_U$ is bijective;

(iii) For every point $a$ of B, the mapping $\varphi_a$ is a bijection of the stalk $\mathscr{F}_a$ onto the stalk $\mathscr{G}_a$.

The implication (i)$\Rightarrow$(ii) is immediate.

(ii)$\Rightarrow$(iii) : consider the commutative diagram (I, p. 51)

where the vertical arrows are the canonical B-isomorphisms. If condition (ii) is satisfied, $E(\varphi |\mathscr{B})$ is a B-isomorphism, hence $E(\varphi )$ is also one. The mappings $\varphi_a$ are deduced from $E(\varphi )$ by passing to the fibres and are therefore bijective.

(iii)$\Rightarrow$(i) : under assumption (iii), the mapping $E(\varphi ): E_{\mathscr{F}}\rightarrow E_{\mathscr{G}}$ is a bijective B-morphism of étalé spaces and therefore is a B-isomorphism (I, p. 30, cor. 2 of prop. 6). Hence the morphism $\widetilde{\varphi}:\widetilde{\mathscr{F}}\rightarrow \widetilde{\mathscr{G}}$ is an isomorphism. Since $\mathscr{F}$ and $\mathscr{G}$ are sheaves, the canonical morphisms $\sigma_{\mathscr{F}}:\mathscr{F}\rightarrow \widetilde{\mathscr{F}}$ and $\sigma_{\mathscr{G}}:\mathscr{G}\rightarrow \widetilde{\mathscr{G}}$ are isomorphisms (corollary 1) and one has $\widetilde{\varphi}\circ \sigma_{\mathscr{F}}=\sigma_{\mathscr{G}}\circ \varphi ($I, p. 54, relation (2)) hence $\varphi$ is an isomorphism.