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Why would they have to keep up with their appraisal? There is no downside to not doing it, other than not being worth as much if they want to sell or IPO, but that would be true even without the equity firm's investment and appraisal. No law says they have to meet the expectations set by a single investor. There is no legal duty requiring a company to maximize profits or shareholder value.
However, even if such a law requiring them to maximize profits existed, it's very reasonable that they would legally continue making the same types of movies that earned them that value in the first place. There couldn't possibly be a requirement for them to change business strategies, else every company would eventually all end up in the same, most profitable industry. They'd all be selling movie theatre popcorn or something.
Because if they don't the PE firm either cancels bonuses for sr management or fires them outright.
You can't beat your boss.
Laws? What the fuck are you talking about? You're right in that their hand isn't forced when making a business transaction such as this, but looking at it through the childish lens of "laws" is odd to say the least.