Why do I keep coughing after getting the flu vaccination?

A few days ago, my family received the flu vaccination.
There is no swelling or warmth at the injection site.

Sometimes you cough repeatedly.

It's not a lingering cough, but a cough like when you have a severe cold, and I hear a coughing sound coming from my lungs.
Once you start coughing, the cough continues for quite a long time.

There are no cold symptoms such as fever, nasal congestion, or sore throat, only a cough.
Could the cause of my ongoing cough be because I received the flu vaccination?