"The image of a category by under a functor is a category."
This is a small one, but it lasted for six months when I was starting in category theory.
A counter example exists in finite case ( counterexample exists, with just 3 objectsonly) , . Even under a connectivity requirement, a small finite counter example exist as wellcounterexample still exists. In fact, it is dead wrong to thing think anything like thatthis holds.