Proposition 5

Let A and B be topological spaces and let $u: A\rightarrow B$ be a continuous mapping.

a) Let $\mathscr{G}_1$ and $\mathscr{G}_2$ be presheaves on B and let $\gamma :\mathscr{G}_1\rightarrow \mathscr{G}_2$ be a morphism of presheaves. Let moreover $\mathscr{F}$ be a sheaf on A and $\varphi :\mathscr{G}_2\rightarrow u_*\mathscr{F}$ a morphism of presheaves. One has the equality

b) Let $\mathscr{F}_1,\mathscr{F}_2$ be sheaves on A and let $\mathscr{G}$ be a presheaf on B. Let $\varphi :\mathscr{F}_1\rightarrow \mathscr{F}_2$ be a morphism of sheaves and let $\gamma :\mathscr{G}\rightarrow$ $u_*\mathscr{F}_1$ be a morphism of presheaves. One has the relation

a) By definition of $\varphi^{\sharp}$ and Remark 2 of I, p. 60, one has

Consequently, $\varphi \circ \gamma =u_*(\varphi^{\sharp}\circ u^*(\gamma ))\circ \beta_{\mathscr{G}_1}$, whence relation (11).

b) By definition of $\gamma^{\sharp}$, one has

whence the announced relation, having regard to the definition of $(u_*(\varphi )\circ \gamma )^{\sharp}$.

Put $\alpha_{\mathscr{F}}$ = Id$^{\sharp}_{u_*(\mathscr{F})}$; it is the unique sheaf morphism $\rho :u^*(u_*(\mathscr{F}))\rightarrow \mathscr{F}$ such that

Id$_{u_*(\mathscr{F})}=u_*(\rho )\circ \beta_{u_*(\mathscr{F})}$.

Relation (11), applied to $\mathscr{G}_2=u_*(\mathscr{F})$ and to the morphism $\varphi =$ Id$_{u_*(\mathscr{F})}$, yields for every presheaf morphism $\gamma :\mathscr{G}\rightarrow u_*(\mathscr{F})$ the factorization

It then follows from proposition 4 that for every sheaf morphism $\psi$ of $u^*(\mathscr{G})$ into $\mathscr{F},\psi^{\flat}$ is the unique morphism $\varphi :\mathscr{G}\rightarrow u_*(\mathscr{F})$ such that $\psi =\alpha_{\mathscr{F}}\circ u^*(\varphi )$.