Which cells engulf antigens by phagocytosis and present fragments of them on their own surfaces for recognition?

macrophages and dendritic cells.

Explanation:

Macrophages and dendritic cells are antigen-presenting cells. They perform a process called antigen presentation. In this process some phagocytes display the engulf parts of the cell to the surface of their cell. Dendritic cells is also known as accessory cells. Dendritic cells act as messengers between the innate and the adaptive immune systems. Macrophages refers to the white blood cells responsible for protection.

Dendritic cells

Explanation:

Dendritic cells are referred to as the antigen presenting cells (APC) of the immune system. They are responsible for the processing of antigens and presenting them on their surface for the action of T cells. These cells are found in tissues that are connected to the external environment and also communicate with other cells of the body.

Dendritic cells are referred to as professional phagocytes like the macrophages and monocytes. They first engulf the antigen by the process called phagocytosis before presenting them on their cell surface. They interact between the adaptive and the innate immune system.