I'm writing a novel with a character named Caitlin who goes by Caty. The story is written in 1st-person. When the character is introduced to the 1st-person narrator as Caty he hears Katy so I originally wrote it with a K. But when I re-read the chapter it seemed like a typo so, even though he's the POV character and heard it with a K, I wrote it with a C. I ended up playing with the confusion, but was I correct in writing it correctly even though the POV character heard it differently?
Here's the section (with sentences left out for brevity):
He laughed. "Cat got your tongue? Or should I say Caty got your tongue."
"What are you talking about?" I asked.
"Caty Brewster." He thrust a thumb in the direction of the girl...
"Hi." I looked up to see the girl putting her belongings down on the empty desk next to mine. "Miss Foster said I could sit here."
I looked at the teacher, but she was busy talking to another student.
"My name's Caitlin," the girl said when she was seated.
"I thought your name was Katy?"
"So you know my name? Wow!" She smiled and her eyes lit up. "My name's Caitlin, but everyone calls me Caty, with a 'C'."
"Oh, I thought- I'm Cory." I smiled and said, "With a 'K'."
"Nah, just kidding. It's with a 'C'."
I did spell it Katy in one place, but in all the other places I spelled it Caty. Does what I did make sense?