Let me first say: I don't love the name "Center of Excellence." This is not about starting a group that has a monopoly on excellence or good ideas with an organization. Just the opposite--this is an entity that helps the organization become more excellent, which includes spotting internal excellence and promoting it.
Regardless of what you call it, an Agile Center of Excellence is meant to be a helpful, consultative group. Not a strict instrument of governance or compliance. While the group may help define mechanisms to promote transparency about product and team health, there is real danger in a COE becoming the internal compliance police.
Digging in further to the idea that an Agile Center of Excellence is a helpful, consultative group that helps an organization become more excellent, the vision of this entity is important. I've found it helpful to use an elevator statement format and Jason Little's strategic change canvas to gain alignment on the group's mission.
Another big challenge in starting an Agile COE is defining success criteria. What are the measurable results you are seeking? Why is this group being established? We often start thinking about the activities or services the COE will provide and how to measure them. I think of those services as the how. Measurements of these activities are our leading measures. I urge you to go deeper: what are the business outcomes wanted that are fostering the COE's genesis? The really important stuff that's probably harder to measure and will take longer to change: increased customer satisfaction, cost savings, more revenue, shorter time to market, etc. What is the reason for agile in the organization?
Why is it so important to define success criteria like this? It hinges on changes from people outside of the Agile Center of Excellence, which feels risky. And it is. Because it means that the Agile Center of Excellence is connected to the organization and must respond to its needs. The COE's success points to the why of the organization's change. I find that it enables--perhaps requires--the Center of Excellence to change, evolve, and pivot its offerings in order to continue helping the organization. It allows for agility by the group, which I think is important for those wishing to further enable agility. How cool would it be to see more Agile Centers of Excellence like that?