Probably. Yet this CDC explanation of mRNA repels me:
Dr Frankenstein also "taught" his monster - how did that work out?
All mRNA - whether from a vaccine or from within the cell - "teaches" the cell, specifically the ribosome, to produce a protein. It's a natural process.
You're literalizing an imprecise metaphor.
Contrary to the popular conspiracy theory, mRNA vaccines do not reprogram, or in any way affect, cells' DNA. The ribosome, where mRNA is used, does not have DNA.