Example 1

Consider a topological space B, a subspace A of B, and denote by $i: A\rightarrow B$ the canonical injection. Let $(E, p)$ and $(E', p')$ be B-spaces. Take for $\mathscr{G}$ the sheaf $\mathscr{M}$or$_B(E; E')$ (I, p. 45, example 4) and for $\mathscr{F}$ the sheaf $\mathscr{M}$or$_A(E_A; E'_A)$. For every open set V of B and every V-morphism $f: E_V\rightarrow E'_V$, put $\varphi_V(f) =f_{V\cap A}$, where $f_{V\cap A}$ is the $(V\cap A)$-morphism of $E_{V\cap A}$ into $E'_{V\cap A}$ induced by $f$. The family $\varphi = (\varphi_V)$ resulting is a sheaf morphism of $\mathscr{G}$ into $i_*\mathscr{F}$. By proposition 4, there exists a unique sheaf morphism $\psi :\mathscr{M}$or$_B(E; E')_A\rightarrow \mathscr{M}$or$_A(E_A; E'_A)$ such that one has

for every open set V of B and every $f\in \mathscr{C}_V(E_V; E'_V)$. The morphism $\psi$ is called the canonical morphism of $\mathscr{M}$or$_B(E; E')_A$ into $\mathscr{M}$or$_A(E_A; E'_A)$.

If A is reduced to a point $a$, this morphism is identified with the stalk morphism at $a$ of the sheaf $\mathscr{M}$or$_B(E; E')$ onto the set $\mathscr{C}(E_a; E'_a)$.

By passing to subsheaves, the morphism $\psi$ induces a canonical morphism of $\mathscr{I}$som$_B(E; E')_A$ into $\mathscr{I}$som$_A(E_A; E'_A)$.