From a specialist role to a product manager

I recently interacted with a professional who moved from specialist / technical work to product / proposition work, and she asked for some high-level advice.

This is what I said... 

1. The business problem(s) you're trying to solve are key, including how you'll measure whether you're making a difference / taking steps towards solving them.  
   
   
2. Relationship and people aspects are crucial, including collaborating with many different functions (like engineering, operations, sales etc). While important, numbers and modelling are much less important in product / change roles than in pricing or other technical roles; and, hence, product roles allow you to build so many more skills.  
   
   
3. Breadth of understanding is very helpful i.e. more generalist than specialist, provided you can call on various specialists as needed. To succeed, you'll need to manage/coordinate across a range of disciplines.  

Do you agree? What else would you add?