The deposit function contains most of the business logic, including minting shares worth of LP tokens in the Validly contract, representing pro-rata ownership of the liquidity pool. The actual token balances are stored in the Sovereign Pool. Source:
The LP can call withdraw to redeem _shares worth of liquidity for the equivalent amount of token0 and token1 balances. Upon burning the LP tokens, token0 and token1 amounts get transferred from the Sovereign Pool into _recipient. Source: