Kakashi would probably first check if Naruto is hungry or needs a diaper change. If that's not the case, he might hold Naruto close and gently rock him while humming a soft tune.
Kakashi would approach Naruto's pranks with a mix of discipline and love. For instance, if Naruto tries to scribble on the walls, Kakashi would stop him gently. He'd then sit Naruto down and show him where he can draw instead, like on a piece of paper. He'd also tell Naruto stories about how important it is to respect their home. When Naruto does something really naughty, like trying to pull the cat's tail, Kakashi would firmly say 'No' but then immediately follow it up with a hug. He wants Naruto to learn right from wrong but also wants to make sure Naruto knows he is loved no matter what.
Kakashi might immediately offer a comforting presence. He could put his hand on Naruto's shoulder and just be there silently, letting Naruto know he's not alone.
Their relationship is like a mentor - mentee bond. Kakashi has trained Naruto, and when he sees Naruto achieve great things, it can be overwhelming. For example, when Naruto masters a new jutsu that Kakashi thought was difficult for him, Kakashi might cry out of joy for Naruto's accomplishment.
Kakashi would probably be very calm at first. He's an experienced ninja who has seen a lot. He'd gently pat Naruto on the back, offering silent comfort. Then he might start to speak softly, asking Naruto what's wrong in a gentle voice.
Naruto could initially feel a great sense of loss and sadness. But as time goes on, he might start to look for new mentors or friends who can support him. For example, he could turn to Iruka-sensei more often or form closer bonds with his fellow classmates like Shikamaru or Choji. He'd rely on his inner strength and the relationships he already has to get through this tough time.
At first, Naruto might isolate himself. He doesn't want to be around others as he is so consumed by his grief. He just sits in his room or goes to a quiet place where he can cry alone and think about Jiraiya.
Well, Kakashi's character becomes softer. Caring for baby Naruto forces him to let down his guard. He was known for his cold exterior, but now he has this little life depending on him. He has to be more nurturing, which is a big change from his usual self. He also becomes more selfless as he puts Naruto's needs above his own. For example, he might sacrifice his own sleep or food to make sure Naruto is okay. This new - found selflessness is a significant shift in his character.
One possible reason could be that Naruto is feeling lonely. Despite having friends, there are times when his past as an orphan and the years of being shunned by the villagers resurfaces, making him feel isolated. Kakashi finding him in such a state might then lead to a heart - to - heart conversation about Naruto's inner struggles.
One possible fanfiction could be that Kakashi stumbles upon Naruto crying alone in the forest. He sits beside Naruto quietly, not saying a word at first. Then he gently puts his hand on Naruto's shoulder and tells him that it's okay to cry, that showing vulnerability doesn't make him weak. Naruto then pours out his heart about his loneliness and how much he misses his parents. Kakashi listens intently and shares some of his own experiences of loss, creating a deep bond between them in that moment.
Kakashi would become more nurturing. In the original, he was a teacher, but as an adoptive father, he'd have to be more affectionate. He might show his face more often to Naruto, as a sign of trust, and be more open about his emotions.