I have just finished my master's course on which I focused on category theory, because I found it a very interesting subject, but my background, as an undergraduate student, was on functional analysis and analytic topology. So I did not arrive to category theory from a "normal" path, i.e logic, set theory, then algebraic topology etc. But I am now very interested on category theory and I would even like to do a PhD on this subject.
My question is, how easy would it be in your opinion, to start learning about things like cohomology theories or abstract homological algebra, by looking at them from a category theoretic point of view, in which I have a stronger background now, than starting, say in homology, from studying singular homology or other simpler cases. Or for another example, if you consider sheaves, would it be useful - in a technical sense- to know and use the definition of a sheaf which uses restrictions of intersections, or does it suffice, for the applications of this idea, to understand the definition as a presheaf satisfying the unique amalgamation on matching families condition? So, I guess what I am asking is, is your ability to generalize concepts, when you are considering research level questions, dependent on your good understanding of the underlying theory in its simple forms? I look, for example at the definition of a product of topological spaces, and see how beautiful it looks to describe using categorical language, but does it feel easier to handle now just because I knew already what this object is?