Cosmo T.D. (which stands for "The Dog") is a golden retriever introduced (i.e. born) in the series premiere of Fuller House.
Cosmo's mother, Comet Jr. Jr. is Steve Hale's dog, and both are descendants of Comet, the original Tanner family dog from Full House. Cosmo is born in the backyard of the Tanner House, where Comet's mother, Minnie was first seen.
His debut birth took place in the Fuller House series premiere, "Our Very First Show, Again," but he isn't formally introduced until Max picks him from Comet Jr. Jr.'s litter and names him three episodes later in "The Not-So-Great Escape."
Just a young puppy in season one, Cosmo is still in the process of being housebroken, as he was caught chewing up the couch in Secrets, Lies and Firetrucks, and still has a tendency to 'potty' in the house (most notably outside Max's door) rather than waiting to go outside.
In War of the Roses, Cosmo displays more bad behavior when he enters the boys' room with one of Ramona's shoes in his mouth. She and Jackson then discover that Cosmo is a "hoarder" after they check under Max's bed and find a pile of items belonging to various family members.
The possibility exists that Cosmo is actually the very same puppy that Max initially named Comet Jr. Jr. Jr., as both are males, and there was no way for Max to keep track of which was which between birth and later picking Cosmo out at Harmon Pet Care. If this is the case, it seems likely that one of the other puppies would be named Comet Jr. Jr. Jr. in order to keep with Steve's unconventional naming tradition.
The dog who portrayed Cosmo, with the same real name, passed away due to surgery complications two weeks before the finale was filmed. His death was announced in December 2019 by the cast, but he had already died in November. When the cast made their final bow, Elias Harger held a picture of Cosmo, to honor him.