`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.