Why do T cells have MHC II receptors?
That's the description in my book which has the same image.
The book is: Janeway's Immunology (9th Edition)
Article discussing MHC Class II expression in humans:
https://doi.org/10.4049/jimmunol.168.2.763
Excerpt from the book:

The processing and presentation of pathogen-derived antigens has two
distinct purposes: inducing the development of armed effector T cells,
and triggering the effector functions of these armed cells at sites of
infection. MHC class I molecules bind peptides that are recognized by
CD8 T cells, and MHC class II molecules bind peptides that are
recognized by CD4 T cells, a pattern of recognition determined by
specific binding of the CD8 or CD4 molecules to the respective MHC
molecules (see Section 4-18). The importance of this specificity of
recognition lies in the different distributions of MHC class I and
class II molecules on cells throughout the body. Nearly all somatic
cells (except red blood cells) express MHC class I molecules.
Consequently, the CD8 T cell is primarily responsible for pathogen
surveillance and cytolysis of somatic cells. Also called cytotoxic T
cells, their function is to kill the cells they recognize. CD8 T cells
are therefore an important mechanism in eliminating sources of new
viral particles and bacteria that live only in the cytosol, and thus
freeing the host from infection.
By contrast, MHC class II molecules
are expressed primarily only on cells of the immune system, and
particularly by dendritic cells, macrophages, and B cells. Thymic
cortical epithelial cells and activated, but not naive, T cells can
express MHC class II molecules, which can also be induced on many
cells in response to the cytokine IFN-γ. Thus, CD4 T cells can
recognize their cognate antigens during their development in the
thymus, on a limited set of ‘professional’ antigen-presenting cells,
and on other somatic cells under specific inflammatory conditions.
Effector CD4 T cells comprise several subsets with different
activities that help eliminate the pathogens. Importantly, naive CD8
and CD4 T cells can become armed effector cells only after
encountering their cognate antigen once it has been processed and
presented by activated dendritic cells.