Snape's first reaction could be one of surprise. He may not expect to be given such a task. He has his own focus on other things in the wizarding world, like his role as a double agent perhaps. But when sent to check on Harry's family, he might start to think about all the implications it could have for his position in the war against Voldemort, as well as his personal history with the Potters.
Well, Snape could be initially very guarded. He would be cautious about approaching Harry's family. Since he has his own secrets and his loyalty to Dumbledore might be in conflict with his personal feelings towards the Potters. He might approach the task with a sense of duty but also a great deal of wariness.
It could start with a lot of mistrust from Harry's family towards Snape. But as Snape shows his true intentions (if he has good ones), they might gradually warm up to him. For example, Harry's mother could see some of the good in Snape that she knew from their childhood days.
In the 'snape is sent to check on harry family' fanfiction, Snape's relationship with the Harry family could be full of twists. At first, he might be there just to do his job. However, as he gets closer to them, he could find himself protecting them in his own way. He might even become a sort of mentor figure to Harry, teaching him things that Dumbledore or others haven't, because he has a unique perspective from being sent to check on them.
One plot twist could be that Snape discovers a hidden secret about Harry's family that changes his whole perception of them. For example, he might find out that there was a prophecy related to Harry that was different from what everyone thought.
This is a very disturbing scenario. In the context of the Harry Potter universe which is generally about magic, friendship, and good vs. evil, such a concept goes completely against the spirit of the original work. Snape would likely be extremely horrified and disgusted, given his complex relationship with Harry. He might see it as a desecration of the students and the world he knows.
His wife could be a former Hogwarts student he met during his own school days. She might be from a different house and they fell in love secretly.
Dumbledore could be involved. He often has his fingers in many pies at Hogwarts, so he might be the one who arranges for Snape to deliver the letter to Harry for some greater good or part of his grand plan. Another possible character could be Hermione. She might notice something strange about Snape delivering the letter and start to investigate, suspecting that there's more to it than meets the eye.
In this fan fiction, Snape's character development is quite remarkable. At first, he might be hesitant and conflicted about saving and adopting Harry. But as time goes on, he becomes more selfless. He has to learn to communicate better with Harry, which is a big change for him. He also has to face his own demons from the past, like his relationship with James Potter. This journey of self - discovery and growth makes his character more three - dimensional and likable in the context of the fan - made story.
Snape's reaction would likely be more complex than others. While others might show immediate relief, Snape might be more guarded. He has a history of animosity with Harry, so he might not show his true feelings right away. He could be more critical, even as he helps Harry, asking him how he got into such a mess in the first place.
Snape might become more empathetic. After seeing Harry's memories, he could understand the pain and pressure Harry has been under. This new understanding might make him more protective of Harry in some way.
He could be secretly touched. Deep down, Snape has a complex personality. Even though he doesn't show it easily, the fact that Hermione, a bright and determined student, gives him a gift might reach his softer side and make him feel a bit of warmth.