When a client application uses the Fabric Gateway client API to create a Gateway connection to a Hyperledger Fabric network, it must supply the client's X.509 certificate and private key. The X.509 certificate is used to authenticate the client to the network, confirming the client's identity and permissions. The private key is used for signing transactions, which is necessary for the network to validate the integrity and origin of the transactions submitted by the client. This combination ensures that both the identity of the client is verified and that transactions cannot be tampered with without detection.