There have been previous assumptions in Arizona that in the court of law with regards to child custody, the mother was assumed to have a stronger role with regards to being a child's caregiver, and therefore would be more likely to gain child custody during divorce proceedings. The courts, however, have always followed the protocol that there is no discrimination on the basis of gender, sexual orientation or race.
Assuming that paternity has been established, this means that all things equal, a father has an equal right to child custody as a mother. With this in mind, there are several other factors that the courts take into consideration when making a ruling regarding a child's custody decision.
In general, the courts will look at each parent's role in the child's upbringing, may ask for the child's own opinion regarding their relationship with their parents and will also look to see if anything else could affect the child's best interests. This could include a history of domestic violence or a criminal record or even a history of drug or alcohol abuse. The courts will also look at the physical and mental condition of each parent as well as the financial situation of each parent during their decision-making process.
The courts, today more than ever, are aware of the importance of protecting fathers' rights, whether it is regarding child custody, paternity, visitation rights or child support. It is equally important for a father to protect his own rights with regards to a child's custody or visitation rights. This may mean speaking with a firm familiar with family law to assure that his best wishes will be taken into consideration during the court's decisions.
Source: the spruce, "Factors Used to Determine Child Custody for Fathers," Debrina Washington, September 28, 2016