Classic/Original:
Sans had just cameback from Grillby's when he remembered that Papyrus was still hanging with Undyne. He was walking home when he heard crying? He walked towards the sound and saw a baby girl with e/c it a pink basket. He picked it up gently picked you up and saw a note it said

Couldn't take care of her so sorry. Her name is Y/n please take good care of her❤

He knew he couldn't leave you here so he took you home with him.

Underfell:
Fell had just got into an argument with his "Boss" again,he was tired and fell asleep at his station. He heard giggle like a little baby,he got pissed and was about to yell at it shut up when he saw and small human baby crawling around his station. He was confused oh how the Hell a baby could even get this far in this cold,his thought was cut off when you laughed while making an "Up" Motion. "Huh how did you get this far?" You giggle before yawning. "Hmmm you must be tired...wanna stay with me and my bro?" You fell asleep on his shoulder." I guess that answered my question...I'm going to call you Y/n" " Yeah that sounds Badass for a name" He teleported you home before realising how was he going to explain this to Boss.

Underswap

Blue was doing his usual "Guard For Humans", His bro however was home most likely sleep. He was doing some puzzles when crying could be heard, He was confused but followed the sound. He saw some monster kids pointing at a basket before one tried to kill you, he doesn't know why but he got angry. He summoned a bone which scared the kids away,he saw that the basket had a baby in it. A human one in fact, he saw a note with the word "Y/n" "Huh what a Magnificent name human!" You start to cry,he didn't know what to do. "Shh P-please human!" You did stop before laughing then end up falling asleep. He found you adorable and decided to take you home.

Error
He had just finished destroying a previous Au when he went back to the "Anti-Void", he was bored until he notice something in a box. He was ready to attack as he got closer but stopped when he saw a Baby? "WhAt? HoW DiD A BaBy GeT HeRe?" He hesitantly picked you up when you woke you,you looked confused but grabbed his finger. He was shocked because he assumed you would have cried apon seeing him but you didn't, you just smiled." I SeE YoUrE NoT AfRaId oF mE HuH....I'Ll KeEp YoU" " BuT I'll ShAlL NaMe YoU....Y/n" He smiled at you as your eyes closed while your small hand was still wrapped around his boney finger.

Ink
He had got into another fight with Error and was tired,he had went to his Au so he could think of more Au's. He stopped when he felt a tug on his shoulder,he turned to see a baby girl in a pink onesie grabbed his shoulder trying to stand up but failed. He grabbed gently and looked around. "Where did you come from?" "Welp I guess you have to stay with me um...Y/n!!" He had little hearts in his eyes as he watched crawl around ignoring the fact that he forgot to make a new Au.

Nightmare
He was his kingdom as usual,he felt more lonely then usual. Then out of Know Where one of his servants had brought him a basket?" Why the Hell did you give me a basket?" "L-look i-inside s-sir" He look inside then when wide-eyed when he saw a beautiful baby girl,you looked at him and grabbed one of his tenacles and sucked on it like a pacifier. He stared at you before slightly smiling which was rare for him, "Hmm I'll name you Y/n, thats it!" He cradled you softly while you slept peacefully in his arms.

Dream
He was looking at dreams as usual when he felt a positive energy,he looked around and saw a baby that didn't even look One yet. He smiled as stars appeared in his eye's,you tried to grab them but he kissed you forehead and held you back to his house. " I'll name you Y/n!"

Fellswap

Sans was walking around Snowdin looking for Papyrus, he asked around but most said "No" or "Haven't Seen him".  He grown annoyed and gave up,he walked around when out of Know Where crying could be heard. He listened to the noise and saw a box,he looked inside and saw a baby. He used his magic to hold it,you laughed. He grinned and decided to take you home " I'll name you y/n.

Horrortale

He was chilling at his stations when he heard crunching in the snow,his eye's shot open with his axe in his hand ready to kill whatever was in front of him. He saw it was nothing? He was confused and looked over to see you. He picked you up and sat you on his stand,you face-planted on the wooden stand making Sans howl of laughter. You giggle to and try and grab his axe,he seemed impressed that you haven't cried yet."You arn't scared are you? I guess not...Ehhh I'll keep you for now...hmmm now for your name how about Y/n? Yeah I like that!"

Cross Sans/Chara
Cross was travelling around Au's as usual,when he entered on that was all white. He saw and brown box and walked towards it, he was shocked to see a baby girl in it "What the-" "Aww she's so cute we should keep her!" "What! B-but I don't know the first thing about taking care of a baby chara!" " I can help you come on please!" He stared at you before sighing,"Fine..." "Yes! So...what should we name her?" " How about Y/n?" "I like it!" "Okay...Y/n it is" he saw that you were still asleep and smiled before leaving the Au.