Forgetful functor from Grp to Set is faithful

Question

I have read that the forgetful functor from Grp to Set is faithful. Part of this means that the map from a group to its underlying set is injective. But I don't see how this is the case.

Let $(G,\cdot)$ be a non-abelian group. Now define an operation $*$ on $G$ by $$ g*h=h.g\quad \forall g,h\in G$$ Then $(G,*)$ is also a group with the same underlying set $G$. Hence the forgetful functor maps both of these groups to $G$ and is not injective. What am I missing?

Answer 1

Nevermind, I misunderstood the definition. The map to the underlying set does not need to be injective. Its the morphisms which must satisfy an injectivity requirement.