John 6:33

ὁ καταβαίνων. That which cometh down. Jesus has not yet identified Himself with the Bread, which is still impersonal, and hence the present participle: contrast John 6:41. There is a clear reference to this passage in the Ignatian Epistles, Romans 7; the whole chapter is impregnated with the Fourth Gospel. see on John 4:10; John 3:8; John 10:9.

τῷ κόσμῳ. See on John 1:10. Not to the Jews only, but to all. We have evidence (the γάρ introduces an argument) that it is the Father who gives the really heavenly Bread, for it is His Bread that quickens the whole human race.