Stroller/ car seat question for flying!

Okay I need help clarifying from mamas who flew with their babies.

I will be flying with my 10 month old in about 2 weeks. We are bringing her stroller and car seat- car seat for the rental car at our destination. However, she will be sitting in me and my husbands lap so she will not be utilizing the car seat.

So here is where I need help clarifying. I KNOW that I can “check the stroller at the gate” and have her stroller at the front of the plane so immediately after deboarding the plane we can use it, BUT can we also gate check the car seat so it is also with the stroller? OR do we have to put the car seat under the plane with the luggage?

Edit: I have also bought a car seat protector bag thing so that if it does need to go under the plane with luggage it is protected from possible damage.