Remark 1

Let A, B, C be topological spaces, let $u: A\rightarrow B,v: B\rightarrow C$ be continuous mappings; put $w=v\circ u$. Let $\mathscr{G}$ be a presheaf on C.

Let U be an open set of C and $s$ a section of $E_{\mathscr{G}}$ over U. Then $\beta_{\mathscr{G}}^v(s)$ is the section $s\circ v$ of $E_{\mathscr{G}}$ over $\overset{-1}{v}(U)$, and $v_*(\beta_{v^*\mathscr{G}}^u)(\beta^v_{\mathscr{G}}(s))$ is the section $s\circ v\circ u=s\circ w$ of $E_{\mathscr{G}}$ over $\overset{-1}{u}(\overset{-1}{v}(U)) =\overset{-1}{w}(U)$.

It follows that $\beta_{\mathscr{G}}^w=v_*(\beta_{v^*\mathscr{G}}^u)\circ \beta_{\mathscr{G}}^v$.