In a left-forbidding grammar, a set of nonterminals is attached to every context-free production, and such a production can rewrite a nonterminal if no symbol from the attached set occurs to the left of the rewritten nonterminal in the sentential form. The present paper discusses left-forbidding cooperating distributed grammar systems that work in the t-mode and have leftforbidding grammars as their components. It demonstrates that with two components, these systems generate the family of recursively enumerable languages.