In team coordination, how to prove your ability individually to the management?

Every part you played in supporting team coordination will be notified by management... for example 

your mail conversation, on time delivery, customer appreciations and your track records will prove your ability to the management .. On our part we have to be perfect in allocated tasks.

Your work is your ability .Hard work & honesty are like fragrance which doesn't require any extra efforts to be proved.

Don't worry, You will be noticed and your touches can easily be identified.  Just do your part of the business and help others to do their part of the business and someone one day will know :).

Working in a better team and success team is a success for everyone

by making every effort and taking the initiative in decision-making

by giving new ideas,and help others in your team

As a team player we are given targets to deliver. Deliver your target on time if possible exceed. Your number will not lie as the number we bring in to the Team is a part of the big number the management needs. I believe that at the end of the day the management will see these numbers and the hard work you have put for it. 

i would like to point out that it is a team work, you have to know if trying to prove your individual skills will give you more credit or create issues in the team. to prove your skills and ability individually you have to make sure that the team members all agree with it. the first thing that will entail is always saying 'in my opinion' so as to make it seem it is your idea, or feeling not what you are imposing on them.

In some hard situations your super power will shine.

Do  not show off and try to proof anything.

It is important to play your part well as a member. Be a team player and a good one at that so that other member look up to you. Do not make the mistake of putting down others in order to raise yourself up, there is something called Karma and it does catch up with you eventually, if have seen many make that mistake and pay the price, short victories are not worth it. Be that person who is known for responsibility and integrity and soon your worth will be recognized by the management, if not, find a new management that can appreciate hard work.