A functor $F\colon \mathcal{C}\to \mathcal{D}$ is surjective on objects if the action on objects
\[ F\colon \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )\to \operatorname {\mathrm{Obj}}\webleft (\mathcal{D}\webright ) \]
of $F$ is surjective.
