In To Kill a Mockingbird is Cal's double life of being a servant for the Finches and a seemingly different person at home justified?
Imagine yourself in Calpurnia's situation as a black woman living in the South in the 1930's. Racism was rampant, her people were constantly discriminated against, poverty was constantly knocking at the door, and you have to try to make a living and navigate through the many social intricacies that are presented to you in your society. Cal was certainly justified in living her life to the best of her ability, to provide for her children and keep herself and her family safe. The way that the question you asked is phrased makes it seem like Cal was a bad person for presenting herself to the Finch family in a way that was pleasing to them, and yet at home fitting in there. There is nothing wrong with her "double life," if you want to call it that. She was not lying, she was not being insincere, she was not pretending; Calpurnia was both of those people--the proper grammar-speaking, rules-following, hard-love mother figure of Scout's childhood, and the more casual, friendly, at ease Cal of her home life. She was "at home" in both places; she did good in both places, she survived and got the job done in both places. That was the way of life for people in her situation at that time period.
If Cal had stepped out of her two separate roles in life and decided that she wanted to be just one identity at all times, imagine the repercussions. She might have lost her job with the Finches, which would keep her from being able to provide for her family. That's not a good option. On the other hand, if she were to abandon her home roots identity and constantly play the role that white society put upon her, she would lose her family, friends, and sense of self and uniqueness separate from being a servant. That too is not a good option.
Calpurnia found a good balance for herself, one that allowed her to make lasting, enduring connections in both situations, to provide for her family, and to give herself stability and happiness. I hope that those thoughts helped; good luck!