The author neatly points out how several recent concepts in the theory of FI-modules, namely the shift functor \(S\), the derivative functor \(D\), and the coinduction construction \(V\mapsto QV\), are related by simple category theoretic ties. The author first recalls a construction \(F\mapsto F^\dagger\), for a functor \(F: \text{C-Mod}\to \text{C'-Mod}\), which, if \(F\) is right exact and preserves direct sums, yields a right adjoint. This result is then applied to exhibit the negative-one shift functor as giving rise to the shift functor, the derivative functor, and the coinduction construction as, respectively, its right adjoint, left adjoint, and extension by it.