The Hermitian adjoint of a Pauli operator

In Pauli theory, the coordinate representation of every one-electron operator is necessarily a \((2 \times 2)\) matrix operator: \[\begin{equation} \require{physics} f^c(\vb{r}) = \begin{pmatrix} f^{c, \alpha \alpha}(\vb{r}) & f^{c, \alpha \beta}(\vb{r}) \\ f^{c, \beta \alpha}(\vb{r}) & f^{c, \beta \beta}(\vb{r}) \end{pmatrix} \thinspace . \end{equation}\] If we define the Hermitian adjoint \(f^{c, \dagger}(\vb{r})\) of this operator analogously to the scalar case as: \[\begin{equation} \matrixel{\phi_P}{f^{c, \dagger}}{\phi_Q} = \matrixel{\phi_Q}{f^c}{\phi_P}^* \thinspace , \end{equation}\] we can derive (by expanding the spinors in their two-component forms) that the Hermitian adjoint of a \((2 \times 2)\) matrix operator is given by the following \((2 \times 2)\) matrix operator: \[\begin{equation} f^{c, \dagger}(\vb{r}) = \begin{pmatrix} f^{c, \alpha \alpha \dagger}(\vb{r}) & f^{c, \beta \alpha \dagger}(\vb{r}) \\ f^{c, \alpha \beta \dagger}(\vb{r}) & f^{c, \beta \beta \dagger}(\vb{r}) \end{pmatrix} \thinspace . \end{equation}\]