USDC and USDC.e equivalent on Uniswap?

I am looking to provide liquidity to the ETH/USDC pool on Arbitrum. Currently, I have ETH and USDC.e in my Arbitrum wallet. The USDC.e if the result of bridging ETH from Ethereum mainnet to Arbitrum.

If I provide liquidity of ETH/USDC.e, will it be included in the ETH/USDC pool (the biggest pool on UniSwap)? I believe that when contributing WETH/USDC, the WETH is considered equivalent to ETH, and therefore included in the ETH/USDC pool. I am wondering whether the same applies for USDC and USDC.e. If not, I would first need to swap USDC.e for USDC before providing liquidity.