Okay, here I am being controversial again. First, I say kuzushi is not important (such heresy!), then I dismiss the value of uchikomi (how dare I?), next I commit blasphemy by saying stretching before training is not only useless, it could be harmful. Now to top it all off, I say that cool-downs after hard training are a waste of time (sacrilegious!).

Anyone who has trained at our club knows that after randori, we don't bother to do any cool-down exercises. We just stop. Some will plump down in front of the floor fan. The reason I don't ask my players to cool down is simple: There's no point to doing so. Confused? Baffled? Bewildered? Okay, some reading material for you.

First off, from the New York Times: Do We Have to Cool Down After Exercise?

Most of us were taught in elementary school gym classes that the body requires a formal period of cooling down after a workout or competition. Instructors told us that by slowing to a jog or otherwise lessening the intensity of the workout, followed by stretching or otherwise transitioning out of physical activity, we would prevent muscle soreness, improve limberness and speed physiological recovery. All of this would allow us to perform better physically the next day than if we hadn’t cooled down. But under scientific scrutiny, none of those beliefs stand up well.

The available data “quite strongly suggest a cool-down does not reduce postexercise soreness,” says Rob Herbert, a senior research fellow at Neuroscience Research Australia and senior author of what is probably the foundational study of cooling down, from 2007.

In that experiment, healthy adults walked backward downhill on a treadmill for 30 minutes, courting sore muscles and curious stares from fellow gymgoers. Some of the volunteers first walked forward for 10 minutes as a warm-up; others did the same afterward, to cool down. Others didn’t warm up or cool down. Two days later, the group that had cooled down was every bit as sore as the control group.

Here's another one from the New York Times: Is the Exercise Cool-Down Really Necessary?

... the cool-down is enshrined in training lore. It’s in physiology textbooks, personal trainers often insist on it, fitness magazines tell you that you must do it —and some exercise equipment at gyms automatically includes it... The problem, says Hirofumi Tanaka, an exercise physiologist at the University of Texas, Austin, is that there is pretty much no science behind the cool-down advice.
As far as muscle soreness goes, cooling down doesn’t do anything to alleviate it, Dr. Tanaka said. And there is no physiological reason why it should. That’s also the conclusion of a study of muscle soreness by South African researchers who asked 52 healthy adults to walk backward downhill on a treadmill for 30 minutes —an exercise that can cause sore leg muscles. The participants were randomly assigned to cool down by walking slowly uphill for 10 minutes or simply to stop exercising. The result, the researchers reported, was that cooling down did nothing to prevent sore muscles.
And muscle tightness? “In a different generation we would have called it an old wives’ tale,” Dr. Foster said. “Now I guess I’d call it an old physiologists’ tale. There are no data to support the idea that a cool-down helps.” But, he added, once again, “it’s an idea we can’t get rid of.” As for Dr. Tanaka, he does not cool down at all. He’s a soccer player and, he says, he sees no particular reason to do anything after exercising other than just stop.

Lastly, here's what The 12-Minute Athlete says:

For as long as there has been research about exercise, cool downs have been around. Coaches, personal trainers, fitness books and magazines insist on it—even the exercise equipment at the gym has an automatic cool down when you select the pre-determined workout option.

Apparently, the thinking behind the cool down is that most people (including trainers and coaches) assume that doing a proper cool down after your workout will assist with recovery and help you feel less sore or keep your muscles from tightening up after a hard workout (whatever that actually means).

The problem? There’s actually no science to back this up. As in, none at all. In fact, one of the first studies actually done on exercise cool downs in 2007 suggested that cooling down had absolutely no impact on muscle pain the next day. Meaning, those who went through the cool-down process after their workout were just as sore as those who did no cool down at all.