Is the butterfly cycle adult eggs catapiller pupa butterfly?

The adult is the butterfly, so those are not 2 different stages. But other than that, yes, those are the stages. In most groups of butterflies, the special term for the pupa is "chrysalis", but the generic term "pupa" applies to that stage in all insects, so you can never go wrong using "pupa". (I'm glad to see you didn't use "cocoon", as that term does not apply to butterflies, only certain moths.) :)