The word "onto" - adjective? According to Oxford English dictionary, the word "onto" is preposition only but I see it used as an adjective in mathematical writings. I think it is grammatically correct to say that "$f$ maps $X$ onto $Y$", but in a very formal writing, is it considered as correct to write that "The map $f:X\to Y$ is onto"?
 A: I do not recommend using "onto" as an adjective as in "The map $f: X \rightarrow Y$ is onto".  I agree that this is conscripting a preposition into service as an adjective in an awkward way, and like many pedantic speakers of English it twangs unpleasantly in my ear a little bit every time I hear it for this reason.  In contrast, "$f$ maps $X$ onto $Y$" is slightly informal, but one can be slightly informal even in serious mathematical writing: I would not bat an eye at encountering it in a published paper.
It seems to me that the wide usage of "one-to-one" and "onto" in student writing is an artifact of the relatively recent (say, in the last 20 years) phenomenon of "transitions classes", i.e., intermediate-level undergraduate classes where students get introduced to logic and mathematical reasoning in an especially careful, deliberate, way.  In particular, the only books in which I see "The map $f: X \rightarrow Y$ is onto" are textbooks for such courses.  
(Well, to be honest, it's a little worse than that.  There is a recent genre of American undergraduate math textbooks which are, as far as I can see, not mathematically innovative in any way whatsoever and whose main pedagogical innovation is the adoption of a new, less literate more informal, friendly style designed to appeal to today's less literate busy students.  Thus I see sentences ending with the word "onto" occasionally in recent algebra, analysis or topology books designed for American undergraduates.  I trust you can tell how I feel about this.)
I really think that the expository reasoning here is no deeper than "We don't want to scare students with unfamiliar, difficult words like injective, surjective and bijective which are not as immediately transparent* as "one-to-one" and "onto".  That might scare them away from the concepts.  Let them learn the fancy words later."
I think that's pretty silly, in particular the idea that university students will find the learning and memorization of three new terms to be a significant hurdle.  (Remember biology class?)  Whenever I teach a course in which such terminology is introduced, I introduce both but make a point of using the latinate words instead.  In later courses I mention the latinate words almost exclusively, but will remind students of the synonyms when necessary.
*: Not as immediately transparent for an anglophone without a strong background in Latin or etymology, I mean.  I speak just enough French to appreciate that it goes over more easily there, and I am enough of a Francophile to like it a little more for that reason.
