To deject is to make someone feel really, really sad. Few things will deject you more than losing your very favorite scarf — the one your grandmother knit — on the subway.
This verb is rarely used these days, so you're most likely to find it in an old book— the adjective dejected is much more common. Both words come from the earlier verb dejecten, "to throw or cast down," from the Latin deicere, "to cast down, destroy, or defeat." The more figurative "make sad" meaning appeared around 1500. If you really want to deject someone, play them some melancholy music when they're already sad about their lost cat.