The competition turns inward when you have a monopoly. Instead of different companies vying for market dominance you have middle managers trying to post the biggest numbers to get that coveted promotion.

Actually, that happens in a Free Market too, but companies have more incentive to tamp down on inefficient behavior if it makes them less competitive. In a monopoly the inefficiency is much harder to prevent.

The word "free" just implies a minimal amount of regulation. Monopolies can and do form in free markets. There's even such a term as "natural monopolies," which are things like utility providers, where it would make little sense to have multiple companies servicing the same market. For example, it would not make a lot of sense municipal water systems servicing the same area, due to infrastructure difficulties.

The term you're looking for is "competitive market." In a perfectly competitive market, monopolies don't form, but, there is also no long term economic profit to be had. That is, the sum of "accounting profit" (what you'd typically think of as the "bottom line" of a company's P&L statement) minus opportunity costs tends to 0 in the long term.

Perfect competition is a highly idealized model that can't happen in practice, but is instructive to analyze in theory. See https://en.wikipedia.org/wiki/Perfect_competition