Should this part move freely? Gears popping out. pics included

[rowbee]

Hi all, please bare with me. I took car to have starter motor changed, on return to me gears keep popping out 2nd and 4th. There was definitely no problem before. Have returned to garage and theyve said there is nothing wrong with gears (100% is it wouldn't stay in 2nd at all when I drove to garage) Had a look under bonnet and spotted this.....
Questions, what is it and should it flap, or be joined (looks like it should be to me)

I can see its not near the starter motor but i know they would have had to disconnect the battery to do the starter motor and battery is right next to this part....is it possible they couls have broken it doing this  or something else needed to replace starter motor?

Thanks in advance

[johnnyroper]

Looks like you are pointing to a gear cable,and yes it should be connected to the gear linkage. Have a look around and locate the part of linkage with a small ball on it and push back on.