I have a question. Is there a definition of restriction of morphism of (locally) ringed space?
Let $(f,f^{\flat}): X \to Y$ bea morphism of ringed spaces ; i.e., $f:X\to Y$ is a continuous map and $f^{\flat} : \mathcal{O}_Y \to f_{*}\mathcal{O}_X$ is a homomorphism of sheaves of rings on $Y$.
Now, let $U \subseteq Y$ be an open set and consider the restriction $f|^{U}_{f^{-1}(U)} : f^{-1}(U) \to U$. Then can we consider a morphism of sheaves $(f|^{U}_{f^{-1}(U)})^{\flat} : \mathcal{O}_U \to (f|^{U}_{f^{-1}(U)})_{*}\mathcal{O}_{f^{-1}(U)}$ ? ; i.e., is there a natural definition for $(f|^{U}_{f^{-1}(U)})^{\flat}$?
Perhaps, for each open $V\subseteq U$, $(f|^{U}_{f^{-1}(U)})^{\flat}_V := f^{\flat}_{V}$ ?
(It seems that it will be good if $\iota^{*}(f^{\flat})= (f|^{U}_{f^{-1}(U)})^{\flat}$, where $\iota : U \hookrightarrow Y $ is the inclusion and $\iota^{*}$ is the inverse image functor.)