Once a pet reaches the Blue rank, they can become a patron of one of the heroes. The list of the heroes available for patronage is limited for each pet.
Patronage increases the stats of the chosen hero in any battle, except the maximum power training battles. What is more, if the pet has reached the Violet rank, the hero will also get the additional pet patronage skill. The pet does not have to participate in the battle for patronage to work.
There is no limit to the number of patrons in a team. You can gather a team with five different patrons, but only one pet can participate in a battle.