Etymology
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cata- 

word-forming element meaning "down, downward," but also "through, on, against, concerning," etc., from Latinized form of Greek kata-, before vowels kat-, from kata "down, downward, down from, down to," from PIE *kmt- "down, with, along" (source also of Hittite kattan (adv.) "below, underneath," katta "along with"). Occasionally in Greek it had senses of "against" (catapult) or "wrongly" (catachresis), also "along, through, over, across, concerning." Also sometimes used as an intensive or with a sense of completion of action (catalogue). Very active in ancient Greek, this prefix is found in English mostly in words borrowed through Latin after c. 1500.

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syn- 
word-forming element meaning "together with, jointly; alike; at the same time," also sometimes completive or intensive, from Greek syn (prep.) "with, together with, along with, in the company of," from PIE *ksun- "with" (source also of Russian so- "with, together," from Old Russian su(n)-). Assimilated to -l-, reduced to sy- before -s- and -z-, and altered to sym- before -b-, -m- and -p-. Since 1970s also with a sense of "synthetic."
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trans- 

word-forming element meaning "across, beyond, through, on the other side of, to go beyond," from Latin trans (prep.) "across, over, beyond," perhaps originally present participle of a verb *trare-, meaning "to cross," from PIE *tra-, variant of root *tere- (2) "cross over, pass through, overcome." In chemical use indicating "a compound in which two characteristic groups are situated on opposite sides of an axis of a molecule" [Flood].

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