What happens if we define a functor $F:C \to D$ to be injective when it is injective on isomorphism classes, or equivalently when it gives an injective fucntion from the objects of the skeleton of $C$ to the skeleton of $D$?

Edit: To be more specific, how does this definition relate to a that of a fully faithful functor?

What happens if we define a functor $F:C \to D$ to be injective when it is injective on isomorphism classes, or equivalently when it gives an injection from the objects of the skeleton of $C$ to the skeleton of $D$?

Edit: To be more specific, how does this definition relate to a that of a fully faithful functor?

What happens if we define a functor $F:C \to D$ to be injective when it is injective on isomorphism classes, or equivalently when it gives an injective fucntion from the objects of the skeleton of $C$ to the skeleton of $D$?

What happens if we define a functor $F:C \to D$ to be injective when it is injective on isomorphism classes, or equivalently when it gives an injective fucntion from the objects of the skeleton of $C$ to the skeleton of $D$?

Edit: To be more specific, how does this definition relate to a that of a fully faithful functor?