I'm a beginner in the subject & my question can be meaningless, so I'm sorry from start if that's the case.
I just don't understand why all of the image f(L) can be a sublattice of L' when f is a homomorphism from L to L'. I learned that all subsets of a lattice don't have to be a sublattice at all. If I don't understand wrong, it's not about the elements of the lattices actually but about the operator differences between them.
So why there is not a possibility of a non-inherited operator of f(L), or a non-sublattice lattice of the L' as f(L)?