In 1981 Seymour proved his famous 6-flow theorem asserting that every 2-edge-connected graph has a nowhere-zero flow in the group ${\mathbb Z}_2 \times {\mathbb Z}_3$ (in fact, he offers two proofs of this result). In this note we give a new short proof of a generalization of this theorem where ${\mathbb Z}_2 \times {\mathbb Z}_3$-valued functions are found subject to certain boundary constraints.