Is collaboration the fast choice for humans? Past studies proposed that cooperation is a behavioural default, based on Response Times (RT) findings. Here we contend that the individual's reckoning of the immediate social environment shapes her predisposition to cooperate and, hence, response latencies. In a social dilemma game, we manipulate the beliefs about the partner's intentions to cooperate and show that they act as a switch that determines cooperation and defection RTs; when the partner's intention to cooperate is perceived as high, cooperation choices are speeded up, while defection is slowed down. Importantly, this social context effect holds across varying expected payoffs, indicating that it modulates behaviour regardless of choices' similarity in monetary terms. Moreover, this pattern is moderated by individual variability in social preferences: Among conditional cooperators, high cooperation beliefs speed up cooperation responses and slow down defection. Among free-riders, defection is always faster and more likely than cooperation, while high cooperation beliefs slow down all decisions. These results shed new light on the conflict of choices account of response latencies, as well as on the intuitive cooperation hypothesis, and can help to correctly interpret and reconcile previous, apparently contradictory results, by considering the role of context in social dilemmas.