There is a crisis for people to monitor the disaster

(Grammar Questions I Couldn't Answer)

This one puzzled me for a long time, but I think I've actually worked it out now.  Let me know what you think.

We were playing a Make Sentences game with the new vocabulary from Unit 2, Impact 4, and we were on slide 4 of the game (see here), and one of the students wrote on the board "There is a crisis for people to monitor the disaster."

My native speaker intuition of course immediately told me that the sentence was wrong, and that it should be, "There is a crisis for the people to monitor."  But why?  Monitor is a transitive verb, so it should have an object after it, right?  So why was my native-speaker intuition telling me that "to monitor the disaster" was wrong in this case?

I wrote the sentence down, and have been thinking about it for several days, and I think I've got it now.  It's because "to monitor" is an infinitive of purpose.  So in the same way that we would say, "I have a pizza to eat" without putting on object after "eat", so we wouldn't put an object after "monitor".  

Or perhaps, more specifically, the object of "monitor" is already in the sentence--it's "a crisis", so we don't need to say it again in the infinitive.

How did I do?  Does that sound right?