A free-forgetful adjunction is a pair of functors between two categories, where one functor (the free functor) creates objects in a more structured category from objects in a simpler category, and the other functor (the forgetful functor) 'forgets' the additional structure when moving back to the simpler category. This relationship highlights how certain constructions can be viewed through the lens of universal properties, allowing us to understand the existence and uniqueness of morphisms in terms of these functors.
congrats on reading the definition of free-forgetful adjunction. now let's actually learn it.