TITLE: How do I check if a functor has a (left/right) adjoint?
QUESTION [85 upvotes]: Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find myself saying, "Hey, cool functor. Wonder if it has an adjoint?" The problem is, I don't know enough category theory to be able to check this for myself, which means I can either run and ask MO or someone else who might know, or give up.
I know a couple of necessary conditions for a functor to have a left/right adjoint. If it doesn't preserve either limits or colimits, for example, I know I can give up. Is there an easy-to-check sufficient condition to know when a functor's half of an adjoint pair? Are there at least heuristics, other than "this looks like it might work?"

REPLY [6 votes]: A special case I have found particularly tractable is when the categories involved are locally presentable.

Theorem: Suppose $\mathcal{C}$ and $\mathcal{D}$ are locally presentable categories, and $F : \mathcal{C} \to \mathcal{D}$ is a functor. Then:

$F$ has a left adjoint if and only if it preserves limits and is accessible
$F$ has a right adjoint if and only if it preserves colimits

(ref: nLab)

A lot of nice categories are locally presentable, and there are enough structure theorems that you can often quickly verify that a category is locally presentable.