B cells and T cells and other cells are constantly interacting as part of the adaptive immune system. The way I look at lymphocytes is that B cells 'see' antigens while T cells 'read' antigen fragments. The B cell receptors bind to the 3 D shape of the antigen or to large fragments of the antigens. T cells don't really respond to the whole antigen but rather to 14-22 amino acid fragments of the antigen that are presented to the T cells in the context of MHC Class II. So they 'read' a sequence of the antigen presented to them but they do not read free antigen or fragments.
As an example of the process (simplified): SARS-CoV-2 spike protein binds to a very small number of naive/mature B cells through the B cell receptor (which is a monomeric IgM) on the surface of the cell. Only a tiny fraction of mature B cells will just so happen to have the right membrane IgM that has some affinity. During B cells development there is recombination of multiple regions on the heavy and light chains to get a specific membrane IgM(BCR). The antigen binding to the BCR can activate the B cells and the BCR bound to the antigen is endocytosed and then broken up into peptides. Some peptides will fit into the MHC Class II molecule on the surface of the cell and now the B cell is also an antigen presenting cell (APC)
The B cell or other APC's (monocyte/macrophages, dendritic cells that randomly collected and broke up the antigen) can now activate a T cell.
The trimolecular complex is the antigen fragment bound to MHC Class II (on teh B cell or other APC) and interacting with the T cell receptor. Other 'costimulatory' proteins interact (CD40/CD40L; CD80 or CD86 interacting with CCTLA-4 and CD28) which is necessary to fully activate the T cell. The CD40/CD40L interactions from these now activated CD4 T cells (with cytokine help also), in turn cause Ig class switching so that these interactions reinforce each other.
Hence ideally, a vaccine will lead to both a humoral (IgG via B cell) and and cellular (T cell) response. An IgA response might also be useful for an infection, like SARS-CoV-2, coming through the mucous membranes.
As a related aside, CAR-T cancer immunotherapy uses genetically engineered T-cells that have a TCR targeting a cancer cell surface protein (like CD19 for B cell lymphoma/leukemia) with an intracellular segment with both stimulatory (CD3) and costimulatory (i.e from CD28) domains.
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