Choice and competition tend to exist with in the free market, but they are not the fundamental premises of the free market.
We know this because choice and competition exist in places where there is no freedom.
For example, an emperor can toss two gladiators in an arena.
On issuing the command "Kill or be killed," the emperor could be treated to an exciting competition.
If a gladiator refuses to fight, the emperor could give the gladiator a choice: "Be tossed in a ring of lions, or be tossed in a ring of vipers."
These are two radically different choices. Both options have serious ups and downs. The situation is not really in keeping with what I would consider the free market.
A good slave owner would give slaves choices and pit slaves against each other in competitions. The slave exercising the limited choices given by the master is still a slave. The competitions exist for the master's benefit and not the slave's.
We know from simple observation that choice and competition are not the foundation of a free society.
The foundation of a free society is freedom and property rights. What a free person experiences goes deeper than simply a defined choice made now and then. A free person is a self-propelled unit that is existentially involved in making his own world.
In a world with free people, we would notice people running around making choices and competing or cooperating with others. The roots of these choices and associations are derived from the freedom and not the other way around.