The Nightmare Police are a pair of bosses with a stupid collective name. Bit will appear in a random uncleared stage you visit after clearing two Maverick stages, and Byte will appear in a random stage after you defeat five Mavericks. They'll appear in Doppler Stage 1 if you don't destroy them both completely, joining together to form a more difficult boss. If you do destroy them, you'll fight a different boss in that stage. To destroy them completely, you'll have to finish them off using one of the weapons they're weak to.
Bit is weak to the Frost Shield and Thunder Triad. If you want to destroy him completely, you'll need to use one of these to deal the final blow. He's fast, and will get faster after his health drops below a certain level. Bit will jump into the air and charge forward when you're both on the same vertical level. Jump into the air to make him dash above your head. You can shoot him from behind while he's dashing to damage him. Get him into a pattern so that you jump when he jumps and he dashes over your head, you dash under him, then turn and jump and shoot him while he's still in the air. Repeat this as much as you can, though he will sometimes stop to do other attacks. You can't prevent this from happening. He can fire a small ring that will move horizontally and vertically around the stage, as though traveling on a grid. The ring will trap you and prevent you from moving if it touches you. It will disappear on its own after a while. Bit can also fire an energy ball that will home in on you. It can't turn very sharply and it will eventually just fly off the screen. After he takes enough damage, he'll start firing two rings or two balls at a time. Bit is fast and does a good bit of damage when he comes in contact with you. He's probably harder than whatever boss is at the end of the stage, so use your sub tanks if you need them.
The best way to deal with him is to shock him with a Triad Thunder as he passes overhead, if you have it. If not, use the Frost Shield as a kind of "trap." It moves slowly, so fire it above his head and he'll jump into it. Then avoid his attack and let him dash to the other side of the screen, and place another Frost Shield above his head and repeat. If you position it correctly, you can hit him under his energy wave that appears when he dashes. Remember to finish him off with the Frost Shield or Triad Thunder if you want him to stay down for good.
You'd think an enemy this big and ugly would be slow, but no. Byte is weak to the Tornado Fang and Ray Splasher, and you'll need to use one of them to deal the killing blow if you don't want him coming back later. Byte uses these nasty magnet mine things on the walls which will push you away from them and make it hard to wall jump. Don't try to continuously wall jump from a wall that has one of these on it. If you start right next to the wall, you can get a good dash jump or regular wall jump with an air dash to avoid Byte's charge. He'll dash at you and do a powerful spinning punch combo. What a butt. If he doesn't hit you, he'll hit the wall and cause it to tremble, knocking you off of it. Normally walls are your best friend in fights with large enemies, but not today. The Tornado Fang might hit him multiple times if he stands in place for a while. Use your sub tanks if you need them.