I disagree with this, and would be very cautious of doing something like this in a business setting.

No matter how nicely you put it, you're going to come across as "I know more than you, I noticed this and you didnt, I can help and you cant". Especially stepping in and offering to coach, without being asked, in a department that isn't yours... I wouldn't like it if I were the other manager, I wouldnt like to see it if I was senior management.

If your environment allows for and encourages a more open structure, where you're confident you're not overstepping your bounds, yes this would be the tact to take. Considering you have to ask HN if this would be overstepping your bounds, just don't. Just stick to your stuff.