struggle, fight, compete, contend
鬭 centers on direct conflict: struggling against an opponent, fighting, competing, or contending for something.
鬭 combines 鬥 (fighting, struggle) with 斲 (cut, chop), suggesting a physical fight or contention. The exact historical development is uncertain, but the component combination fits the meaning of struggle and combat.
The outer 鬥 already suggests two figures in conflict. Inside, 斲 looks like a tool for cutting or chopping. Picture two opponents locked in a struggle, each trying to chop the other down.
For トウ, imagine the fighters are toe-to-toe in a fierce struggle: toe -> トウ, and they are locked in combat.