I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook.
The skillset I am least satisfied with is the application of the Category Theory to Algebraic Geometry. The thing is that Category Theory itself, within the context of the course, seems quite trivial. The example solutions make not much use of it, and its use was marginal in the homework.
However, exam papers of previous years appear to require some relatively advanced and dense juggling of things like fibre products, exactness of functors, and limits -- and at least more advanced than I am prepared for. I seem to lack the required intuition to use these formalisms effectively. It seems that there is more emphasis on Category Theory in those exam problems than in Hartshorne.
I have tried doing problems in Category Theory as well as going over with pen and paper of more advanced problems relevant to the issue, but I don't appear to make much progress because although I may understand how a solution of a more advanced problem works technically, my intuition is not substantially improved.
I am actually a quite bright person (measured by practically everything else, anyway), and I think that there must be an optimal strategy to address the issue. Also, I think that the problem I am experiencing is encountered by many students studying Algebraic Geometry.
So I would like to ask for advice on a strategy to approach this problem. Perhaps there are specific references and/or problem sets that can be recommended.