A first connector part (110) 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 a side of the circular opening opposite to the valve chamber. The circumferential wall forms a concave receptacle (118) for receiving a right circular truncated cone of the second connector part, the concave receptacle having a shape of a truncated cone and facing away from the valve chamber, the circular opening being located in a center of the concave receptacle. The second connector part (150) for establishing the fluid connection with the first connector part comprises the right circular truncated cone (151) for being received in the concave receptacle of the first connector part; a recess (153) at a tip of the right circular truncated cone for actuating the valve member of the first connector part; and a fluid feed conduit (152). The fluid feed conduit has one or more outlets (154) opening at least partially toward a shell surface of the right circular truncated cone (151).