The team knows Peter as Tony's personal intern but doesn't know he is Spiderman.
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Peter had just finished patrol when he realized his heating system was down. Figuring he should bring it to Tony to fix it with him. So he swung to the tower, entering through the spider window Tony had installed just for him. He didn't think anyone who would there so he called out after taking off his mask.
"Mr. Stark? Are you here? My suits heater is broken again."
He walked into the living room, expecting Tony to be lounging around on the couch working. What he didn't respect was to see the Black Widow, or has he called her Nat, sitting on the sofa staring at him.
"Ummm, I can explain?"
"Was that a question or an answer Peter? Or should I call you Spidey?"
"Please do not tell the team! They can not know this yet. They would freak. Please, Nat."
"Fine. But, I am calling you my baby spider now."
Peter grinned and jumped onto the sofa, curling up into Natasha's side.
"I can get used to that, mama spider"