Harold Finch
Detective Carter was the best cop he ever knew. She helped him even when he didn't deserve it, and she protected John when he couldn't.
He may never have approved of John's fascination with her, but he grew to understand why, and kept his thoughts to himself, because his thoughts didn't matter. She made John happy, even if John didn't realise it.
By the time Carter arrested John the first time, Harold had received his number multiple times, but since, his number never came up again.
And even if John didn't know it, Harold cared for him very much.
John Reese
Joss Carter became his world a very long time ago, and when she died, he almost lost the will to live again. She saved him the first time, but now she couldn't.
He'd always go to her if she asked, and even if she didn't. He would do anything for her, to repay her for saving him, but she was stubborn. She liked to do things alone.
He loved her and her stubbornness, and he hated every moment he knew she needed help and he couldn't do anything.
When he held her as she died, he had lied to her, to himself, that she would be able to live, that he wouldn't have to check in on Taylor, and now he had a duty to fulfil.
He would check in on Taylor between cases, but he wouldn't ever allow himself to become a large part of his life like he had done with Joss, so he could save Taylor from meeting the same end.
He couldn't save her, so he'll try his hardest to save her son.
Lionel Fusco
After Carter was murdered, all Lionel wanted to do was hunt down Simmons, and kill him, yet he couldn't bring himself to actually do it.
He thought of his son, and he thought of Carter's son. Carter's death made him think about reality, how he was almost killed, and how it was always a close call.
It wasn't ever anyone's fault, yet the guilt lingered. He felt guilty for her death, for not being there to take the bullet for her.
She was a good person, better than him, and he planned to show John that you never stop fighting. He just kept fighting for what is right, and hope it is worth it in the end, because that's all that counts.
Sameen Shaw
Shaw didn't particularly care for Carter, but she never really cared for anyone, so it went without saying.
She was a sociopath, incapable of caring for others, but Shaw did care, enough to leave behind a body count while looking for Simmons.
He wasn't going to get away with it, because Carter meant a lot to the others, especially John, and she couldn't bear the fact that a killer was lingering.
She hated killing, but she was good at it, because she didn't care.
She just knew it would be for the best, and that he would never cost an innocent life again.
Root
The Machine cared for Carter. Root never understood why, but she accepted it, knowing she wasn't to question the machine that was always right.
The Machine had almost begged her to save Carter, but she couldn't, so Root knew that Carter wouldn't make it. The Machine accepted that after a while, and when quiet, allowing Root to stop arguing with her.
They were both terribly stubborn, but they both agreed on one thing; Carter wasn't supposed to die.
She never told Harold because she knew he would never get there in time, and she would carry that guilt forever.
But her death allowed HR to finally be taken down, and even though there was nothing worth killing for, Root found comfort in the fact that there was something worth dying for, and that's what Carter did.