I am trying to see if we can interpret an \\(n\\) by \\(m\\) real matrix \\(\mathbf{A}\\) as a functor so that its conjugate transpose \\(\mathbf{A}^*\\) is the right adjoint in the categorical sense and use it to understand the meaning of naturality. However, I am too caught up with the "traditional way" that \\(\mathbf{A}\\) is a morphism from \\(\mathbb{R}^m\\) to \\(\mathbb{R}^n\\) and have a hard time to rethink \\(\mathbf{A}\\) as a functor between suitable categories.