Does the mother retain the right to custody of her daughter after remarrying, or does custody transfer to the father, especially given that the father shows no interest in the custody of his daughter?

Islam is keen on children and their good upbringing, and on prioritizing the child's interest above all else. The default is that the mother has a greater right to custody of her young children after divorce, unless she remarries. This is derived from the noble Sunnah (definition) of the Prophet. The mother is given precedence in custody and breastfeeding due to her knowledge and ability to raise the child and her compassion towards him. The right of custody falls from the father or mother if either of them neglects the child or falls short in his best interest, and custody transfers to whoever looks after the child's interests and preserves him. If the girl reaches seven years of age, the father has a greater right to her for her protection and preservation, unless he is incapacitated or negligent. Parents should cooperate for the child's benefit, and in case of dispute, the matter should be referred to the ruler, or the parents should agree to arbitrate a man of intellect and religion to rule according to what he deems to be in the child's best interest.