Kan extension

Summary

A Kan extension extends a functor $F \colon \mathcal{A} \to \mathcal{C}$ along another functor $K \colon \mathcal{A} \to \mathcal{B}$ to a functor on $\mathcal{B}$, in the best possible way. There is a left Kan extension, which is universal among functors receiving a natural transformation from $F$, and a dual right Kan extension. Limits, colimits and adjoint functors are all special cases. This is why Kan extensions are often called the most basic universal construction of category theory.