No! Many founders ask if they should have some sort of agreement with their co-founders to set out each one’s rights and responsibilities when they start a company. This type of agreement is not needed for most startups, and adding another agreement into the mix after incorporation is likely to cause confusion, conflict, and higher legal fees down the line. A typical set of startup incorporation documents includes a certificate of incorporation, bylaws, board consents and stock purchase agreements, among others. These documents set up basic corporate governance responsibilities for founders. That baseline is usually all you need to get started.
Starting a company requires trust, flexibility, and faith. If you and your co-founders feel that you have to adopt an additional enforceable contract to clarify your roles and responsibilities beyond designating titles, it is worth reconsidering whether you will make a strong team. No matter how comprehensive your co-founder agreement is (and how much you paid a lawyer to draft it), it will never govern every fundamental aspect of a relationship among founders. Even worse, the terms of your co-founders agreement could wind up conflicting with traditional good corporate governance practices established in your incorporation documents. The company will grow and pivot; roles will change; people will come and go. The co-founders agreement which seemed like a good idea at the time of incorporation could become irrelevant, or even worse, sow the seeds for litigation if a founder exits on bad terms.
Of course, there are situations where stockholders or executives want to clarify their relationships with the company and each other. Those most often arise with more mature companies and seasoned executives where a significant amount of money (rather than just equity) is at stake. If you are interested in adopting one, ask a lawyer.