You searched for: “inter mural
inter-mural, intermural; intramural
inter-mural, intermural (in tuhr MYOOR uhl) (adjective)
Describing the involvement of participants between two or more educational institutions, athletic clubs, or other groups; literally, "between the walls": There were friendly inter-mural debates scheduled among the rival schools.
intramural (in truh MYOOR uhl) (adjective)
Concerning something which exists or is carried on within the bounds of an institution; especially, a school, athletic club, or a group; literally, "within the walls": Kevin signed up to play intramural soccer at his school.

The winner of the intramural sports competition at each of the universities then played an inter-mural game to determine the overall championship.