B-lymphocytes produce antibodies during an immune response. different antibodies act differently and there are some antibodies called opsonins which attach themselves onto an antigen and mark them for phagocytes to recognize them because phagocytes have antibody receptors on their surfaces. Macrophages simply cut up a pathogen to display the antigens on their surface which are then recognized by neutrophils. However they can be stimulated by cytokines(hormones produced by T-lymphocytes) to carry out phagocytosis more vigorously. Neutrophils make up 60% of White blood cells so they are the main cells responsible for phagocytosis but like I said, if cytokines are produced by T-helper lymphocytes, also macrophages can carry out phagocytosis. If you need further explanation just ask
