Right adjoints preserve limits

Remarks

Let $(L, \lambda)$ be a limit cone of a diagram $K \colon J \to \mathcal{D}$. For an object $c$, cones from $c$ over $G K$ match, by the adjunction, with cones from $F c$ over $K$. These match with morphisms $F c \to L$, and so with morphisms $c \to G L$. So $(G L, G \lambda)$ is a limit cone.

Example: the forgetful functor from groups to sets has a left adjoint, the free group. So the underlying set of a product or an equalizer of groups is the product or the equalizer of the underlying sets. The free group functor preserves coproducts: the free group on $X \sqcup Y$ is the free product of the free groups on $X$ and on $Y$.

The theorem gives a test for non-existence. The forgetful functor from groups to sets does not send the initial trivial group to the initial empty set, so it has no right adjoint.

Going deeper

For a student

This is the most used property of adjoint functors. It explains many familiar facts at once, such as that tensor products distribute over direct sums, since $- \otimes M$ is left adjoint to $\Hom(M, -)$. It is also the necessary half of the adjoint functor theorems. Those theorems show that, under size conditions, preserving limits is enough for a functor to have a left adjoint.