When you are talking about (lost and found) in hotels What is the section that must be based follow-up Housekeeping or human resource ?

If you are talking about lost and found for a hotel I think it the housekeeping staff should turn the items into the front desk staff and they should be tasked with following up with the guests as they would have the information who stayed there and their contact information. They can contact the guests and have them positively ID the items and arrange to get the items back to the guest.

First hygiene and follow-up by housekeeping and secondly the speed of response and the services provided

Well talking about lost and found the only department to be contacted is Housekeeping

Housekeeping Department is the responsible for the lost and found item and follow as per company policy to call the guest or not and keep in lost and found store.

Housekeeping Department is the first responsible for the lost & found items and it should be hand over to Front Desk for guest inquiries if ever and it will be surrendered to Security Department for keeping the item/s.

immediately report to the supervisor

House Keeping personnel is responsible for the lost and found.  

Housekeeping Department is the responsible for the lost in found item/s in hotels then the housekeeping needs to surrender the said item/s to security department.

2 I have a work in the hotel and we have no problem on this issue We have lost and found items found in a special stool safe

You must take the lost or found thing to the responsible person there so that he may find and give the owner so that other day may come back to the are because there is enough trustworthy and much security for the guests

housekeeping departement is the responsible about the lost and found