I really want to breed my two Aurleons but my female is an adolescent. How do I make her an adult? Just battling? Or waiting?

you have to wait. there is an item that can speed up the growth of an adolescent called a Magical Pie that will add (if i remember correctly) 144 points to their maturity, they're listed as "philters" in the item broker. :)

pets age over time.
If a pet is genetically tested or made via an essence, you can see their exact age in maturity hours, which is the number that the site uses to track what life stage a sylesti is at. It is slightly different from their real-time age, as each stage of life can be sped along towards Adulthood in one way or another

If we look at Diaval you can see this collection of numbers in her stats:
Birthday: 11-13-2022 07:06:32AM
Age: 1 Day
Maturity: Adolescent (246/648)

The last set of numbers, in the brackets, are what are relevant here, the (246/648)

246 is her current age-
a sylesti obtained via breeding starts at 0, in the eggling stage
egglings hatch at 72 mh,
hatchlings reach adolescence at 216mh- this is also the age that Generated pets like starters begin life at.
adolescents hit adulthood at 648 (the second number in the brackets)

as such, with her current age being (246/648), you can do some quick maths and work out she has 402 hours left to adulthood (bit less than 17 days)

As mentioned by Yurei, you can speed up the growth of an Adolescent sylesti via the Magical Pie item, which adds 144mh to an adolescent pet. A max of two can be used per pet, as there's a cooldown before you can use another after use.

(egglings and hatchlings can be sped in their growth via correct nurtures- 6 for an eggling skips 1mh, while it's 4 for a hatchling, to balance out that hatchlings also get +1 to their lowest stat for every 8 correct nurtures)