You have basically the right idea, despite what some category theory boosters on this site will say. It's a very useful language for algebraic topology, algebra

You have basically the right idea, despite what some category theory boosters on this site will say. It's a very useful language for algebraic topology, algebraic geometry, and related fields, to the point where working in those subjects without its terminology would be exceedingly clumsy. Otherwise, its value is inversely proportional to your distance to those areas, more or less.