Adjoint functors

Summary

Two functors $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ are adjoint when maps $F c \to d$ match naturally with maps $c \to G d$. Then $F$ is the left adjoint and $G$ the right adjoint. Adjunctions are everywhere in mathematics: free constructions, products, quotients, tensor and hom, and much more. Each adjunction gives a monad, and right adjoints preserve limits.