Most "hard results" in any subfield of math cannot be derived from pure categorical nonsense like adjunctions. At some point, you have to roll up your sleeve an

Most "hard results" in any subfield of math cannot be derived from pure categorical nonsense like adjunctions. At some point, you have to roll up your sleeve and do the work. How would you prove the tychonoff theorem purely with adjunctions? How about the representation theorem of finitely presented abelian groups? And on and on and on...