Everyone will agree with me that there are many levels of abstraction category theory can be introduced at. It makes no sense to start undergraduate math courses with a formal approach to category theory, I don't think anyone would argue the opposite. It makes very little sense either to postpone it to higher algebra classes of late undergrad at best or, as happens in many places, in graduate studies.
Category theory is above all a formalism, a way to frame our understanding. It has been a more and more prevalent facet of my thinking that a good notation does half the work of solving a problem, just as formulating a question properly does. Why then not start hinting towards such formulation early ? While teaching low level courses, I always have, or make a point to ensure that most of my class knows what a function is. While doing so I draw little blobs representing sets and big arrows representing a function. Then as I talk I keep presenting functions as a processes, or relations. Together with a fun example (I usually use a "friends and beer" variation) it helps them structure the knowledge they are presented with. It makes it easier to have them understand that one cannot just "add" functions by writing a plus sign in between since functions are (visually) not the same entity as numbers. It is I believe our duty to frame things as early as possible in a way that structures knowledge in the student's mind. To make another reference to food, it is better to have widely spread malleable foundations of rudimental cooking than of an elite of highly qualified cooks (of course it's best to have both).
Moreover I would like to point out that this formalism is urgently needed in other areas of science. As a physicist by training I cannot overstate the importance of category theory in areas of science other than mathematics. And even after a MS in theoretical and mathematical physics, "functors" and "categories" were frightening words that were reserved to Jedi Masters. I am but saddened by that state of things. About everything in physics deals with processes and change and yet there seems to be very little push to spread the categorical lingua. Relativity screams category theory (equivalent views of the world in different frames yet non identical), the standard model's soul is categorical (groups, tensor structures of representations, etc...). Why should we wait so long to plant these seeds ? Why not let them germ throughout the student's curricula.
In conclusion while it is dysfunctional to force feed students categories (why teach an intensive Japanese course to someone that just wants to make suchi ?), it is criminal to keep it, to its core, our little secret. I believe we need to join forces to move very basic categorical formalism to bigger circles, sans tambours ni trompettes (without fanfare), and without bells and whistles.