Should the non biological parent of a child, entitled to full custody?

*only if the biological parent is an unfit parent
GREYS ANATOMY SPOILER
Example scenario;
Sophias biological father is deceased. The biological mother (Callie) is in a relationship with Arizona. Sophia has a combination of all 3 of their last names as her own.
Fast forward to now, Callie and Arizona are divorced and both moved on, they get along just fine and are still good friends. Callie is in a serious relationship with Penny and is moving 6 hours away to New York as Penny got accepted into some research program over there. Arizona, saw how happy the 3 of them were together and decided to go see a lawyer about obtaining full custody is Sophia despite Callie being a great parent.
Bare in mind that they will only be living in New York for a year and Callie has said to Arizona that she can see Sophia every weekend/they'll make it work. 
Should Arizona be able to do this? Should she be entitled to the same rights as the biological parents? 

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