The people who bought Obamacare during the law's first open enrollment period are largely pretty satisfied customers, a big new study from the Commonwealth Fund shows.

Most adults with new coverage have used it to go to the doctor; and about 80 percent say they're satisfied with their purchase.

That Obamacare's new enrollees would be happy with their new coverage wasn't necessarily a given. Many enrolled in plans with narrow networks, which limit which doctors and hospitals patients can see, restrictions that can frustrate subscribers. Others gained coverage through Medicaid, which pays doctors less than most private insurance plans — and has fewer providers to see patients as a result.

But about six months into the insurance expansion, Obamacare's buyers seem to be, on balance, not as much frustrated with these types of challenges as they are happy with having insurance coverage to begin with.

There are 9.5 million fewer uninsured adults after open enrollment

Before Obamacare's open enrollment, an estimated 20 percent of adults between 19 and 64 reported lacking health insurance coverage. In this Commonwealth Fund survey, conducted between April and June of this year, that number had fallen to 15 percent. That works out to about 9.5 million adults gaining coverage (that figure does not include children).

The biggest gains in coverage were among young adults, whose uninsured rate fell by 28 percent. These people gained coverage both by signing up for private insurance and enrolling in the public Medicaid program.

A majority of those with new coverage say they're better off

Fifty-eight percent of those who signed up for Obamacare — either for Medicaid or private insurance — said that they were better off than they were before than had their new insurance plan.

Those who were mostly likely to say they were better off were Medicaid enrollees; 67 percent reported that they were doing better with the new insurance coverage.

The Commonwealth Fund also shows that 70 percent of people who have used their coverage said they were better off, compared to 41 percent of those who had not yet taken a trip to the doctor or filled a prescription.

Four out of five enrollees are satisfied with the new insurance

Seventy-eight percent of the people surveyed by the Commonwealth Fund reported being satisfied with their insurance. Groups with the highest satisfiaction were those who had enrolled in Medicaid (where 84 percent of enrollees reported being "somewhat satisfied" or "very satisfied" and lower-income Americans who earn less than 250 percent of the poverty line (about $30,000 for an individual).

And three out of five enrollees have used their new insurance — most who couldn't afford care before

This was arguably the Commonwealth Fund's most surprising finding: how quickly Obamacare enrollees started using their health insurance. More than half (60 percent) have already used their coverage to pay for health care services.

The survey homed in on the people who answered yes, they had used some kind of health care services, and asked whether they could have been able to access or afford this care before getting their new health insurance plan. Sixty-two percent said no, they would not.

Some doctors are out of reach — but appointments seem relatively easy to get

About four in ten Obamacare enrollees reported having signed up for Obamacare say they chose a narrow network plan, with lower premiums — but also fewer doctors and hospitals.

This explains why just over one third (37 percent) say that all of the doctors they wanted were part of their insurance plan's network. It's possible that number could be an underestimate, as 39 percent said they weren't sure which doctors were and weren't included.

At the same time, subscribers who did see a doctor generally reported being able to schedule appointments pretty quickly. Most were able to get an appointment within two weeks.

Overall, the Commonwealth Fund survey is pretty positive news for Obamacare. It suggests that the law is accomplishing two of its major policy goals: increasing the number of Americans with health coverage and improving access to health care services.