No Beta. Wrote this just to get it out of my brain before I forgot. Sorry in advance for the lack of detail.
For this story, Katara is approximately 19-20. She lived through the war, but hasn't married Aang. I haven't watched ATLA for a few years but just finished all of the Naruto manga and I decided Kakashi needed someone so... here's that story.
Kakashi spread the scroll out before him on the forest floor. Since the end of the war and the loss of his sharingan, he had decided to train in new ways. The youth of the village were strong. He was proud of that. He was also concerned... his position as hokage meant he should be stronger, as well. At the very least equal to his former students.
He carefully read through the origin scroll, expecting more stories about Kaguya and her sons. If he could tap into some ancient type of power before all the different jutsu were broken out between clans, anything that would help him gain better strength over his chakra natures, he was sure he could become stronger. He already well surpassed most of his students and those of the village, but without the sharingan, he wanted his own power. Kakashi genuinely cared for Konoha and if they had trusted him to lead them, he wouldn't let them down. He never wanted to lose students or friends ever again... he had to be able to take care of any and all threats to the village himself. If not, what good was he as hokage?
"Bend?" He mused, reading aloud. "Four elements? There are more than 4 natures." Had even the elements been broken down through the generations? According to the scroll, his favorite lightening chakra nature belonged within fire. This made no sense.
"If this was written by a drunk Jiraiya to mess with future hokage..." his eyes passed over a summoning spell. Yep, Jiraiya. Nobody else would be so obvious with their prank. And the old man had certainly enjoyed writing stories.
"If I perform this and a toad appears," Kakashi shook his head. There were four summoning spells listed and he decided to start with water, a chakra nature he was comfortable with. Should a toad appear, he would politely apologize and return the scroll to the hokage private library. If, on the other hand, he was able to manifest or materialize a token such as a stone or trinket that aided his training, it would certainly be worth the effort.
Kakashi took a deep breath and followed the hand gestures as they were spelled out. Water, Ice, Bear, Sea and more complicated gestures he wasn't familiar with. When he had finished, he took a deep breath and waited. With a quick puff of smoke, he saw immediate results in front of him.
As the smoke cleared, he saw no toad or token, but a young woman. Her bright blue eyes searched the woods in a panicked state until they landed on him. He noted her blue garment, a dress, and her long brown hair accentuated by two strands pulled to the front, a necklace that hugged her throat gently, the image of it resting against her beautiful brown skin... he saw the ice spike racing towards him before he could process it. Kakashi jumped to the side, narrowly avoiding injury.
He held his hands up in defense. "Please, I'm not here to harm you."
"Who are you? Why am I here?" Her voice was nowhere near ladylike. He had mistaken her appearance for that of a gentle female. She had quickly proven him wrong.
"My name is Kakashi. I am the current hokage of Konoha. What village are you from?"
"Konoha?" She repeated.
"Yes." Kakashi nodded slowly, not wanting to startle her.
"In what nation is Konoha? What is the nearest major city? Ba Sing Se? I am Katara from the southern water tribe." She stood up straight, relaxing from her defensive pose and smiled kindly at the shinobi. "I'm not sure how I got here, but I've seen stranger things happen." When he kept his hands raised, she continued. "I mean you no harm either. I think when you've been labeled a criminal and outlaw and been on the run for years from the fire nation, you learn to attack first." When his eyes widened, she quickly added "not that I am a criminal. Not anymore... it's a very long story, Kakashi. For now, could you tell me which way is toward Ba Sing Se? I was meeting a friend there for an important meeting. United Nations type meeting."
Now Kakashi was the one confused. "Water tribe? I'm not familiar with a water tribe."
Being that there were only two, Katara wasn't sure who this man was that wasn't familiar with water tribes. "What nation is Konoha?" She repeated her earlier question.
He hesitated based on her previous explanation, "the... Uh... the land of fire." His eyes creased in an innocent smile and he jumped back as a whip of water seemingly came from nowhere to lash at him. "Take me to Zuko. Now. Or I won't miss again. Despite the peace between the nations, I know many in the Fire Nation don't agree with the treaties."
Kakashi was once again confused. "I don't believe we come from the same world, Katara. If not for the last Great War, I wouldn't have thought other worlds to be entirely possible. As I've seen firsthand that they are, please allow me to explain... I may have accidentally summoned you."
She was right. She didn't miss. Water hit the side of his face and Kakashi jumped back again. "Look, Katara, I meant it when I said I meant you no harm. We aren't talking about the same land of fire. There is no Zuko here. From your water jutsu, should I assume you're from the village hidden in the mist?"
Katara glared at him and Kakashi felt she was more of a threat than he initially realized. "I am Katara of the southern water tribe, a water bender. If you don't tell me how to get back home, Aang will come looking for me."
"I'm not trying to keep you here. By all means, leave."
"You brought me here, now send me back."
"I can't... I didn't write the summoning spell. I don't know how to reverse it."
"Who does?"
"Well... considering the scroll summoned a lovey woman such as yourself, I'll have to assume it wasn't Jiraiya. Which means... I've no idea."
Katara sighed and sat on the ground. She may have had a quick temper, but it didn't seem to last long.
"Why did you bring me here?" Her voice was calm, quiet. She was looking off into the forest, but he could tell she was still on her guard.
Kakashi suddenly felt sheepish, inadvertently stealing a young woman from her home to better hone his shinobi skills. "I needed help with my water nature chakra."
"You're a water bender?"
"Not quite. Not what you are. I..." he struggled with an explanation. Instead, he decided to simply show her. A quick hand sign and a flash of lightening had Katara's eyes widening.
"You are fire nation."
"No, I can also," another sign and he made a poor job of manipulating the earth in front of them, "use other elements. Lightening comes most naturally to me, like water to you, I suppose."
"What are you doing with your hands?"
"The symbols create a jutsu based on the users power and intention. Your water bending, as you call it, is similar. It's stronger, though. You can manipulate any and all water, I take it? No signs needed?"
She didn't answer, instead posing her own question, "if I help you, will you get me home? I've helped an avatar train before. Maybe that's why you brought me here...?"
Kakashi held out a hand to help her up. "It's getting late. We should head back to Konoha so you can rest. I promise to help you get back home."
Katara nodded, seemingly comfortable with trusting an Avatar, whatever that was.