Comment by slightwinder

Without knowing anything specific of your situation, it reads like this is meant to replace existing solutions? So start with documentation. Collect what everyone is using, how they are using it, why they are using it, and especially find the special cases. A library has to be a common ground for everyone, but with legacies you always have a million small details which will harm you with a million little cuts. So find them early and plan accordingly.

And add as much documentation as possible, but also guides for migration, especially for the special cases. One of the most important parts of such a multi-team-project is communication, so build good connections to everyone, have them involve without annoying them and support them as much as possible for the later migration, if they will be one. And honor that everyone has their own preference, flavors and circumstances, so be open enough in whatever you design to give them some space and flexibility in how they use your components.