and fits into a commutative diagram: where is the natural transformation with for any object of . The functor L is often written . The use of the word "the" (as in "the left Kan extension") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique up to unique isomorphism. In this case, that means that (for left Kan extensions) if are two left Kan extensions of along , and are the corresponding transformations, then there exists a unique isomorphism of functors such that the second diagram above commutes. Likewise for right Kan extensions. Properties Kan extensions as (co)limits Suppose and are two functors. If A is small and C is cocomplete, then there exists a left Kan extension of along , defined at each object b of B by where the colimit is taken over the comma category , where is the constant functor. Dually, if A is small and C is complete, then right Kan extensions along exist, and can be computed as the limit over the comma category . Kan extensions as (co)ends Suppose and are two functors such that for all objects m and m of M and all objects c of C, the copowers exist in A. Then the functor T has a left Kan extension L along K, which is such that, for every object c of C, when the above coend exists for every object c of C. Dually, right Kan extensions can be computed by the end formula Limits as Kan extensions The limit of a functor can be expressed as a Kan extension by where is the unique functor from to (the category with one object and one arrow, a terminal object in ). The colimit of can be expressed similarly by Adjoints as Kan extensions A functor possesses a left adjoint if and only if the right Kan extension of along exists and is preserved by . In this case, a left adjoint is given by and this Kan extension is even preserved by any functor whatsoever, i.e. is an absolute Kan extension. Dually, a right adjoint exists if and only if the left Kan extension of the identity along exists and is preserved by . Applications The codensity monad of a functor is a right Kan extension of G along itself. References External links Model independent proof of colimit