Correcting the captain, or offering input to which the captain should listen? Sometimes it's fine line, and sometimes it's a thick, ugly one.
I've flown with Captains in the past who wouldn't listen, didn't care to receive or solicit input, and who made me feel about an inch high. I've flown with others who were just the opposite. I've flown with First officers who were so incompetent that I dared not let them touch the controls, and two whom I questioned their basic credentials. I've also had first officers for whom I wrote letters to the Chief Pilot recommending them for upgrade. Skilled, well qualified individuals. All kinds, in other words.
As a Captain, I'm the PIC. I consider my crew very valueable, but you have to remember, it is not a democracy. There should always be an air of consideration and cooperation, but it is not a democracy.
Conversely, when I'm not the PIC, I try hard to stay within my bounds. A good first officer won't neglect to bring anything to the Captain's attention that he feels needs to be brought to the captains attention...but a good first officer doesn't necessary strike out to correct the captain. Except...
The first words when the first officer does offer correction, such as being low on speed, off glideslope, etc, are "Thank you, correcting." I firmly believe that "thank you," time permitting, is one of the most correct responses because it expresses grattitude and respect for the first officer's performance of his or her duty. To fault or interrrupt or hold back that duty would be a great shortcoming on the part of the captain.
If instead, the first officer has come from a background which doesn't use the centerline during taxi and takeoff, for example, he or she would be inappropriate to try to correct the captain who does use the centerline...this is a technique issue. In a corporate aircraft, the captain who carries adequate fuel reseves but forgoes taking on a full bag of gas in favor of cheaper fuel down the line, makes a personal, company oriented decision. The first officer who tries to correct the captain there may be making a mistake...only in the case of safety related items, or items which the captain may have forgotten, should the F/O try to correct the captain. The rest of the time, he should respectfully suggest.
One might just as well interchange the terms pilot-flying with pilot-not-flying in some of the cases described above, though the F/O should always remember that he or she is NOT the captain, and that it is NOT a democracy.