Once the project has been approved by the third-party-funding organization, the organization (or a commissioned project management agency) sends a grant notice to the applicant(s). Synonyms for this are terms such as "approval" (e.g. DFG) or "grant agreement" (DAAD).
The grant notice is legally binding and regulates the relationship between the funder and the grantee.
In most cases, the university administration receives the complete grant notice (including attachments) and the department performing the work receives a copy.