suppose $X$ is a smooth variety, with a finite open covering $U_i, i=1,....,k$. $F$ and $G$ are coherent sheaves on $X$, and $G$ is locally free.

Suppose on each $U_i$, there is a sheaf morphism $h_i: F_{U_i}-->G_{U_i}$.

Can this be extended on X ?

i started by taking push-forward, for each i: $$(h_i)_*: (h_i)_*F_{U_i} ---> (h_i)_*G_{U_i} = (h_i)_*O_{U_i} \otimes G. $$ Restrict the map $(h_i)_*$ on the subsheaf $F$ to get a morphism into right hand side.

Using the covering $U_i$ of $X$, can we conclude that the image of restriction, is in $\cap_i (h_i)_*\cal {O_{U_i}} \otimes G$, (which is equal to $G$) ?