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.