 Such a bijection is called an adjunction between \(\mathbf{Set}\) and \(\mathbf{C}\). Here \(F: \mathbf{Set}\rightarrow \mathbf{C}\) and \(U: \mathbf{C}\rightarrow \mathbf{Set}\) are respectively the left and right adjoints of this adjunction; we say that \(F\) is left adjoint to (or of) \(U\) and \(U\) right adjoint to \(F\).