**Exercise 6.1.6**

Consider the theorem at the end of chapter six:

**Theorem**:
Let be a small category and a category with all limits of shape .
Then defines a functor which is right-adjoint to the diagonal functor .

We wish interpret this theorem in the special case that and is a group.

First note that is the category of -sets – that is, groups equipped with a -action . In this scenario, the diagonal functor takes a set to , where is now equipped with the trivial -action where for all and (alternatively, for all ).

To find out what does, we first need to consider what constitutes a cone over a -set.
Given a -set , a cone consists of a set and a single map such that for all .
To rephrase, that means that for all and .
This means that every is stabilized by the whole group .
So a cone is a pair where must take each element to a -invariant element.
So a *universal* cone (limit cone) can be computed by the usual construction giving arbitrary limits in .
We find that the limit is precisely the subset of consisting of -invariant elements.

We are finally ready to interpret the adjunction. The adjunction means that for any set and -set . Indeed, one can observe that a -equivariant map from to can only map onto -invariant elements of , so of course such a map is determined completely by a map , which is precisely the subset of -invariant elements.

The dual of the preceding theorem says that colimits are *left*-adjoint to the diagonal functor.
In the present context, let’s interpet the functor.
Consider some -set .
We can use the general construction for colimits in set and find that the colimit is , where is the precisely the equivalence relation induced by the -action (orbits correspond to equivalence classes).
So, the adjunction says that .
Indeed, its easy to see that for a -equivariant map , all members of an orbit must map on to the same element, so of course such a map is determined by a mapping from the orbits of to .