Green’s Dictionary of Slang

straight-arrow adv.

(US) honestly.

1960		G. Swarthout Where the Boys Are 30: ‘TV,’ I said, ‘straight-arrow. How come you invited me?’.
1970		J. Wambaugh New Centurions 187: ‘We got a lieutenant that‘s very explicit about nice legal pinches [...] The boss says we do police work straight arrow’.