Co-ops can be capitalistic and are capable of functioning under capitalism, but they would also work much the same under any market economy. Decisions and would be profits are democratized/socialized.

Capitalism is a system of capital accumulation with the people who own the means of production hiring workers to operate them. Co-ops are a market economy, but they're demonstrably not capitalism because capital is distributed fairly amongst the workers doing the work. Learn the difference between markets and capitalism.