## § Limit is right adjoint to diagonal

Suppose a category C possesses all small limits. This means that for any index category J and functor F: J -> C, the limit lim F:C exists in C. We wish to show that the functor const: C -> (J -> C) given by const(c) = \j. c has a right adjoint lim: (F -> C) -> C which produces the limit of a diagram. So we are saying that const |- lim. So we need to provide a morphism (const c -> diag) -> (c -> lim diag). A morphism const c: J -> C -> diag: J -> C is a natural transformation between the const c functor and the diag functor. This is, by definition, a cone with apex c. However, every cone factors through the limit cone of the diagram diag. Thus, we get a morphism (c -> lim diag), from the fact that the cone with apex c factors through the cone with apex lim diag, as lim diag is the universal cone. This establishes that limit is right adjoint to diag. From this, can we get a cheap proof that right adjoints preserve limits ? Suppose L: C -> D, R: D -> C are adjoint L |- R. Now, consider limits in D. This can be considered by taking the category (J -> D). We get an adjunction const: D -> (J -> D) |- lim: (J -> D) -> C.
C<-g-D <-lim-   (J -> D)
C    D          (J -> D)
C-f->D -const-> (J -> D)

composing gives us:
C <-g- D <-lim-   (J -> D) <-f._ -  (J -> C)
C      D          (J -> D)          (J -> C)
C -f-> D -const-> (J -> D)  -g._ -> (J -> C)

I'm not sure how to proceed further, but I feel that it must be possible to proceed! I lack the technology, unfortunately, to make this go through.