Data Mesh on AWS Lake Formation

Hi, I'm building a data mesh in AWS Lake Formation. The idea is to have 4 accounts: account 0: main account account 1: central data governance account 2: data producer account 3: data consumer

I have been looking for information about how to implement the mesh in AWS and I'm following some tutorials that are very similar to what I'm doing:

https://catalog.us-east-1.prod.workshops.aws/workshops/78572df7-d2ee-4f78-b698-7cafdb55135d/en-US/lakeformation-basics/cross-account-data-mesh

https://aws.amazon.com/blogs/big-data/design-a-data-mesh-architecture-using-aws-lake-formation-and-aws-glue/

https://aws.amazon.com/blogs/big-data/build-a-data-sharing-workflow-with-aws-lake-formation-for-your-data-mesh/

However, after having created the bucket and uploaded some csv data to it (in the producer account), I don't know if I have to register first to the glue catalog in the producer account or I just do it in the lake formation like it says here: https://catalog.us-east-1.prod.workshops.aws/workshops/78572df7-d2ee-4f78-b698-7cafdb55135d/en-US/lakeformation-basics/databases (is this dependant on if one uses glue permissions or lake formation permissions in lake formation configuration?)

Indeed I have done it first the database and the table in glue and then when I go to lake formation in the database and table sections the database and table created from glue appear there without doing anything. Even if I disable there the options: "Use only IAM access control for new databases" "Use only IAM access control for new tables in new databases" both the database and table appear there

do you know if glue and lake formations share the data catalog? and I'm doing it correctly?

thanks, John