My intuition for that, and you can tell me if its wrong.
The normal explanation for Word2Vec is 2 weight matrices, so the formula looks like this: (One_hot_input x W1) x W2, which is then softmaxed.
W1 then is the matrix that contain our focus embedding from, but if we only evaluate specific words on the target side, then W2 are actually our context embeddings, and the normal multiplication then is focus_w x context_w.