D denotes the homomorphism poset of finite directed graphs. An antichain duality is a pair F, D of antichains of D such that (F→) ∪ (→D) = D is a partition. A generalized duality pair in D is an antichain duality F, D with finite F and D. We give a simplified proof of the Foniok-Nešetřil-Tardif theorem for the special case D, which gave full description of the generalized duality pairs in D. Although there are many antichain dualities F, D with infinite D and F, we can show that there is no antichain duality F, D with finite F and infinite D.