GloVe vector representation homomorphism question

Question

In the paper GloVe: Global Vectors for Word Representation, there is this part (bottom of third page) I don't understand:

I understand what groups and homomorphisms are. What I don't understand is what requiring $ F $ to be a homomorphism between $ (\mathbb{R},+) $ and $ (\mathbb{R}_{>0},\times) $ has to do with making $ F $ symmetrical in $ w $ and $ \tilde{w}_k $.

Am I misunderstanding something? We want $ F $ to be unchanged if we either interchange $ w_i $ and $ \tilde{w}_k $ OR interchange $ w_j $ and $ \tilde{w}_k $, right? Is this the only way to achieve the symmetry between $ w $ and $ \tilde{w}_k $?