We introduce the concept of a functor category to better understand natural transformations. Then we study category equivalence. Most of us notice that intrinsic linear algebra, namely linear algebra independent of basis, and matrix linear algebra are essentially the same. Both can be formulated as a category of finite-dimensional linear spaces and linear transformations over a field such as \({\mathbb {R}}\) , \({\mathbb {C}}\) , and other variety of fields. Intrinsic linear algebras are far wider than matrix linear algebras. These two categories are clearly not isomorphic, yet we have a feeling that they are essentially the same. The concept of category equivalence is invented to tackle this dilemma.