When you have cross company projects and need to have a workflow of assigning different stake holders automatically without the need to manually creating automation or assign people every time a task moves on you can create a repository of your teams Matrix and connect it to multiple project board.
The challenge is how to keep automations, restrictions and reporting working for you with connected boards and mirrored columns.
Many of VLOOKUP app - a very popular monday marketplace app can help. Here is a video demo showing how to setup a repository boards that is mapping your teams across boards.
Here is a small quick workaround using VLOOKUP