I have this problem, I gave my follower a daedric bow and daedric sword, a dragon flies over and he pulls out a standard longbow
There is a problem with the bows that followers use, such the choice of bow seems to be based upon a combination of the damage the bow does, and the type of arrow (and sometimes the type of melee weapon they are also holding). However, this theory only seems to work up to a certain damange, after which, the follower will just use their default bow regardless (which seems to be the 'hunting' bow). The only weapon combination that I have found that seems to work consistently, is to give them a 'forsworn' bow. That particular bow can be upgraded at a workbench and enchanted, and then given to a follower with any type of arrow you give them. Also, with this bow, you can give them any melee weapon and they will have no problem switching between it. Note that some followers will only use a bow when an enemy is in the air, and always switch back to a melee weapon when the enemy is on the ground (e.g. housecarl followers do this). Other followers (like Annekke Crag-jumper) will tend to always use the bow (preferring to attack at long range), and only switch to a melee weapon when the enemy is very close to them. This differences allow you choose a follower that best matches your play style.