I am really surprised that you are asking about this as if this is something that still needs to be proved. Read the UN materials on this issue. They are very informative.
Basically this is something you have to research yourself because the number of scientific articles are so numerous that I don't even know where to start. When omicron came it became obvious that most people – vaccinated or not – will get it. Vaccines are only protecting people by reducing disease severity and deaths (about 90% protection).
Nevertheless, many politicians haven't updated their knowledge therefore insists on stupid mandates.
For more layman reading this article is good: https://www.theguardian.com/world/2021/aug/10/delta-variant-...