'vaguely-established' originates from the combination of 'vague' and 'establish'. 'Vague' comes from Latin 'vagus', meaning 'wandering, uncertain', and 'establish' comes from Latin 'stabilire', meaning 'to make firm'.

'Vague' changed from the Old French word 'vague', and 'establish' evolved from the Old French 'establir', eventually forming the modern English term 'vaguely-established'.

Initially, 'vague' meant 'wandering or uncertain', and 'establish' meant 'to make firm'. Together, they evolved to describe something not clearly defined or firmly set.