Enables DNA-binding transcription repressor activity, RNA polymerase II-specific; RNA polymerase II transcription regulatory region sequence-specific DNA binding activity; and SMAD binding activity. Acts upstream of or within several processes, including embryonic morphogenesis; nervous system development; and regionalization. Located in chromatin. Is expressed in several structures, including nervous system; oocyte; pericardial region; presumptive diencephalon; and ventral mesoderm. Orthologous to human FOXH1 (forkhead box H1).