 Universal property of an inverse image sheaf

Every f-morphism of sheaves from $\mathcal{G}$ to $\mathcal{F}$ factors uniquely through the canonical f-morphism from $\mathcal{G}$ to its inverse image sheaf. A local inverse-image section is represented by sections of $\mathcal{G}$ on image neighbourhoods; their prescribed images glue in $\mathcal{F}$.