Oliver was not standoffish towards Supergirl per se.

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Archived from the IMDb Discussion Forums — Marvel/DC

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fartinglikethedickens — 9 years ago(November 30, 2016 12:37 PM)

Anyone who's ever seen him before knows that such is Oliver's personna.
Mick was much more of a douche towards her, although funny as hell. That line about "seeing everyone's bits"

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ngrosskurth — 9 years ago(November 30, 2016 02:08 PM)

Was taken right from Smallville.

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michaetolandl2001 — 9 years ago(November 30, 2016 04:48 PM)

Yeah. Barry reassures her that Oliver isnt much of a "Trusting Right Away" type. But its good that Digg trusts her instantly.
"Why you so Super?" heat visions S
Digg- "im convinced"