Definition 6.21.9. Suppose that $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. Suppose that $\mathcal{F}$ is a sheaf on $X$, $\mathcal{G}$ is a sheaf on $Y$, and $\mathcal{H}$ is a sheaf on $Z$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map. Let $\psi : \mathcal{H} \to \mathcal{G}$ be an $g$-map. The *composition of $\varphi $ and $\psi $* is the $(g \circ f)$-map $\varphi \circ \psi $ defined by the commutativity of the diagrams

\[ \xymatrix{ \mathcal{H}(W) \ar[rr]_{(\varphi \circ \psi )_ W} \ar[rd]_{\psi _ W} & & \mathcal{F}(f^{-1}g^{-1}W) \\ & \mathcal{G}(g^{-1}W) \ar[ru]_{\varphi _{g^{-1}W}} } \]

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