From English hosting, it means "to host". In feline breeding, a typical contract between a breeder and a private individual relating to the management of a cat is called this. The breeder delivers a cat to a "guest" who will look after it in all respects even though he is not the owner. The reasons that push towards this contractual form are many. I'll show you mine.

When I saw the first litter of Penelope (daughter of Crissie and Bono) the desire to bring home some of the DNA of my ancestors (Crissie and Bono were now sterilized) was very strong. However, two young men (Rhum and Serafino) in the same apartment would have been difficult to manage, and the female under the same roof with her brother and "uncle" was too much at risk of inbreeding. So Serafino went to stay with Chiara and Safira with Pia and Alessia.

I will remain the owner of both of them as long as they are in my breeding program, then I will make the change of ownership and they will definitely be "them".

Other cases may foresee the achievement of some objectives: for example the breeder has other cats to bring to the expo and needs a "guest / exhibitor" to help the cat make the exhibition career. So at the end of the championship, once the titles in the goal are obtained, the ownership is transferred.

For the host it can be a way to realize a dream that would otherwise remain so, because many breeders do not sell breeding puppies as long as they are an active part of the breeding project. It can be a way to start a kennel while keeping the starting costs low, or simply have a cat without paying for it in full, giving up property rights for some time.  

There may be different forms of Hosting, both with economic agreements and with forms of partial "use" (mountain or puppies) but here the possibility of choice is truly vast and the two interested parties will have to agree and put in writing as much as possible.