A functor $F\colon \mathcal{C}\to \mathcal{D}$ is bijective on objects[1] if the action on objects
\[ F\colon \text{Obj}\webleft (\mathcal{C}\webright )\to \text{Obj}\webleft (\mathcal{D}\webright ) \]
of $F$ is a bijection.
Footnotes
[1] Further Terminology: Also called a bo functor.
