When I was replacing the lock cylinder, it got jammed and I had to take the lining apart.
In doing so, the part shown in the picture fell out and now it does not lock properly.
Can you please tell me what part this is and can I find a solution? What to do with it?
It is the coupling of the previous cylinder (which is usually inside the cam) that must be broken to open the lock, leaving the rest of the cylinder inside the lock.
However, if the current lock, after replacing the cylinder, does not close properly, it may be because when the lock was opened when the cylinder was split, they deformed the lock case (in which case the previous cylinder will also give some failure), or because the new cylinder does not have the cam that the cylinder itself needs.
Generally, in profile cylinders, we have a long cam and/or a short cam, but in some cylinders with a long cam, the cam shape (dimensions), does not allow some locks to work properly.
Also, sometimes, due to age and wear on the inside of the lock, when we install a new cam, they don't work properly until the cam begins to wear due to use.
If you provide pictures of the lock, cylinder and current key, we can more accurately surmise the possible problem as well as ways to fix it.
Apparently the previous cylinder was broken and that piece remained inside the lock and would prevent it from giving its corresponding turns, now that you have taken it out you put the new cylinder and it should not give problems.
So you have to ghange the cylinder once again.