'ethically-run' is a modern compound formed in English from 'ethically' (the adverb of 'ethical') + 'run' (the verb meaning 'to manage or operate'). 'ethical' ultimately comes from Greek 'ethikos' via Latin and French; '-ly' is an adverbial suffix from Old English/Old Germanic.

'ethical' derives from Greek 'ethikos' (relating to character or custom) → Latin/French forms 'ethicus'/'ethique' → English 'ethical'. 'run' comes from Old English 'rinnan' (to flow, run) and developed through Middle English into the modern verb 'run', which additionally acquired the sense 'to manage' (as in 'run a business'). The compound 'ethically-run' is a straightforward modern English formation combining the adverb and verb/participle.

Individually, 'ethical' originally meant 'relating to moral character' and has kept that sense; 'run' originally meant 'to move quickly' or 'to flow' and broadened to include 'to operate/manage'. The compound now specifically means 'operated in a morally responsible way.'