An example from a pedagogical perspective: the category of based vector spaces, i.e. vector spaces with a chosen basis, and morphisms matrices, is equivalent to the category of vector spaces by the functor which forgets the basis. Similarly: the category of homotopy types with a choice of CW complexes realizing the type, and morphisms homotopy classes of continuous maps. Generally, this happens any time a choice of extra structure is always possible and doesn’t change the morphisms, which of course is just rephrasing the request that the forgetful functor be essentially surjective and fully faithful!