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Why they don't leave their photographs in a frame on the Bounty were the same:
- It could easily be lost
- Or damaged, especially as they were all kept busy with their work as Ninja
- So they could always have it on them, no matter where they were (in the First Realm, for example)
Zane
The photograph he found at the treehouse of him and Doctor Julien which he brought back and kept on his person always. It was the only photo of the two of them, they had never thought to take another and there wasn't a digital copy of it either.
Somewhere along the way (he suspected when Aspeera had struck him with her staff), it must have been lost somewhere in the Never Realm.
Too bad.
Lloyd
He had lost the photograph of him and Garmadon (when he was cured of the Devourer's venom) after he had been chucked through three walls of solid concrete by the fully-evil version of Garmadon. There wasn't a digital copy of his photograph either.
As he looked back now, there were so many times he wished he could capture the moment in a photograph but was never able to. Lloyd wondered what it would be like to have all the important moments in his life captured in various photographs, it would probably be pretty interesting.
Kai & Nya
The photograph that the two of them both shared was a small, simple one of them and their parents, outside their shop. They didn't quite know what to think when Ray and Maya were revealed to be alive. Their photograph now served no purpose.
Both were so young when the two had disappeared (or taken, as they now knew) and that photo had been hanging on the wall of Four Weapons at the time. Nya had been barely older than three at the time of their disappearance, so she didn't remember as much as he did about them and usually had the photograph.
Now that Ray and Maya were both back in their lives, they didn't really know what to do.
Jay
When he made his final wish and everything had reset, he had lost one of his two photographs, or rather, it was never taken in the first place.
The other, though, did exist and was still in his room in the Bounty. It was of his birth parents, whom he never got and never will ever meet. Jay still wasn't exactly sure how he ended up living with Ed and Edna. Unlike the other Ninja, he had another copy of the remaining photograph, it was in his room at his foster parents' junkyard.
Cole
His photograph was of his mother, simple as that. He still remembered her but not nearly as much as he wanted to. She had died when he was ten, only three years before he had met Wu on top of that mountain. Not long after her funeral, Lou burnt all (or so he thought) photographs that had her in them.
Later that same night, Cole had saved one, small, slightly-burnt photo from atop a bush.
Each and every one of their photographs are of high personal value to them and they all involved family members.