Yes, the King explains

They gave Artoria a few minutes to recover. Taiga thankfully had the natural skill of being an exceptionally deep sleeper, so as Shirou and the harem gathered around Artoria, they weren't interrupted. After Artoria recovers, her and zou look at the group to explain their past with Kiritsugu. It was a long story, given it was effectively the tale of the Fourth Holy Grail War.

Turns out Artoria had low opinions of Kiritsugu, not keeping the disdain from her voice when going into his 'less than honorable' tactics, or how he neglected both Irisviel and Ilya. Zou pitched in with information on his background and the Einzberns, having been one of those who created the grail collaborating with the latter and researching the former.

"So... That was a lot to take in..." Shirou said, having learned a lot more about the last grail war than he had expected. Kiritsugu hadn't told him much about his past. Mostly stuck to how he had wanted to save people and that dream had become impossible for him.

"So... Ilya's the little girl that attacked us after we left the church?" Shirou asked, Saber nodding in return. "I can't believe all that happened to the little girl..."

"Well, after all these years, she's not little anymore... not in the way you might be thinking." Zou interjects, drawing their focus on him. "She was 12 when the grail war was held a decade ago, which means..."

"That she's 22?" Shirou looked at Zou in partial disbelief, the former monster nodding.

"That would be the Homunculus traits in her." Artoria confirmed.

Shirou thought about it for a minute, before deciding it could wait. He grabbed the sex note and began writing. "We'll get to that in a moment. Saber, I think it's about time we did something." The king of knights blanched, having been caught off guard, but Shirou then turned the book around. "Thank you for sharing that with me. I think it's time I made it official.

On the page, Artoria could see her name, and... that she was part of the harem.