Uniswap is a set of smart contracts deployed to the Ethereum network, which means the entirety of this process takes place on-chain.

Overview
Uniswap’s unique architecture sheds the concept of a limit order book entirely. Strangely, market makers no longer specify price when providing liquidity. Instead, they merely supply the funds and Uniswap takes cares of the rest.

https://medium.com/scalar-capital/uniswap-a-unique-exchange-f4ef44f807bf