You are correct. It was both GKC's writing and 'arguments' with Tolkein, who was his collegue at Oxford at the time.
Thanks for the clarification.
Certainly Tolkien was instrumental, but I've long been under the impression their relationship blossomed some time after Lewis's embrace of Christianity. Doug Gresham has written some interesting vignettes of his trips to the pub with "Jack" to meet with Tolkien, where he was party to many of their discussions and exchanges.
A bit of interesting trivia: In the recent Lion film, there is the scene where the children are sitting indoors at the professor's estate due to the rain and war news is coming over the radio. The voice on the radio is Doug Gresham, Lewis's step-son.