At the Gym is where you upgrade the stats of your Follower cards.
The cost of attempting an upgrade is 100TP (earned in the Classroom or, after the Valentine's Day Patch, through dungeons, fights, or practice) and a second copy of the card you're attempting to upgrade (both of which are consumed upon attempting the upgrade). Addtional upgrades require more (unupgraded) copies of the follower to attempt (all of which are consumed). If the required number of extra copies is two or more, you can pay tokens to reduce the by one number of copies required for one attempt.
Although the odds of success are unknown, one can to use the 'Advanced Gym', which doubles the rate of success in exchange for 100 Tokens per use.
Because of the low chance of success for both methods, it's advised only to use the Advanced Gym while attempting to upgrade a card of significant value, and to only use the regular Gym upgrades for Common cards which take little to replace.
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Upon a successful upgrade, your Follower will gain stats depending on its rarity.
Common and Uncommon cards usually get the following:
STLV1 = Base stat +1 Attack (Approxiamately 70-80% chance, Requires 1 copy of the card)
STLV2 = Base stat +1 Attack + 1 Stamina (Approxiamately 30-40% chance, Requires 2 copy of the card)
STLV3 = Base stat +2 Attack (Approxiamately 10-20% chance, Requires 3 copy of the card)
STLV4 = Base stat +2 Attack +1 Stamina (Approxiamately 5-10% chance, Requires 4 copy of the card)
STLV5 = Base stat +2 Attack +1 Defense + 1 Stamina (Approxiamately 1-5% chance, Requires 5 copy of the card)
Event and Rare followers gain +1 Attack + 1 Stamina on STLV 1.