Theoretically that's why the vaccine is supposed to be so effective, because it presents via MHC molecules to the T-cell, which induces T-cell memory, while a traditional vaccine does not have this advantage. The spike protein has to be manufactured via mRNA in order for the MHC presentation to happen according to what I've looked at. This is the same thing that happens in a real infection. I say theoretically because I haven't seen any actual data that empirically shows how the vaccine is behaving once taken up by the vaccinated person's cells, but this is the most plausible explanation for how well it works.