The Inner Circle was nominally complete in 1884. To say the Met and the District both operated the service is a bit of a misnomer - they were fierce rivals and regularly disrupted each other out of spite, e.g. chaining rollingstock to the track. (I could go on about this but Jago Hazzard has many more examples in his videos.)
They were both forced by Act of Parliament to operate the line but in a completely ridiculous way. The Met provided the clockwise service and the District provided the anti-clockwise service. If you bought a District ticket from, say, Paddington to Baker Street you had to go the long way around!