A first disclosed connector part (11 0) for establishing a fluid connection with a second connector part comprises a valve (111) with a valve seat (112), the valve seat comprising a valve chamber (116), and a circular opening with a circumferential sealing lip (114)； a valve member (113), the valve member being provided in the valve chamber, and being able to sealingly close the circular opening of the valve seat when being pushed against the circular opening； and a resilient element (115) that subjects the valve member to a bias force pushing the valve member against the circular opening of the valve seat. The valve member further comprises a circumferential wall around the circular opening, located on the side of the circular opening opposite to the valve chamber. The circumferential wall forms a concave receptacle (18) for receiving a corresponding connection cone of a second connector part, the concave receptacle having the shape of a truncated cone and facing away from the valve chamber, the circular opening being located in the centre of the concave receptacle. A second disclosed second connector part ( 1 50) for establishing a fluid connection with the first connector part comprises a right circular truncated cone (151) for being received in a corresponding concave receptacle of a first connector part； a recess (153) at the tip of the truncated cone for actuating a valve member of a first connector part； and a fluid feed conduit (152). The fluid feed conduit has one or more outlets (154) opening at least partially toward the shell surface of the cone (151).
