Remark

The canonical morphism $\sigma_{\mathscr{G}}:\mathscr{G}\rightarrow \widetilde{\mathscr{G}}$ of I, p. 53 corresponds, in terms of étalé spaces, to the isomorphism $j_{\mathscr{G}}$ of proposition 2 of I, p. 53. It follows that $u^*(\sigma_{\mathscr{G}})$ is an isomorphism. In particular, if A = B and $u=$ Id$_A$, the presheaf $u^*(\mathscr{F})$ is the sheaf $\widetilde{\mathscr{F}}$.