The amount of power the other two Consuls had was very limited during the Consulate. Sieyès, one of the architects of the Brumaire coup, had initially conceived of a layered and more balanced form of executive power that would thread the needle between a weak executive and collective leadership. In this task, Sieyès failed. Partially, this failure was because the experience of both the Revolution and the Terror necessitated a strong figure who would allegedly be above factional politics, but mostly because Sieyès's "sword"- Napoleon- demanded a stronger executive.

In theory, the other two Consuls were invested with some degree of political power and authority. The Article 40 of the Constitution of Year VIII had provisions for the replacement of the First Consul by the other two and this made them a potential check against the power of the First Consul. The Constitution had placed limits on the First Consul's ability to command French armies which theoretically weakened the Consul's ability to act independently. But Article 41 gave the First Consul broad and very ill-defined executive powers and relegated the other two Consuls as advisors to French executive authority. The result was the two Consuls became more akin to aides for the French executive rather than checks on the First Consul's power.

The two men selected by Bonaparte as his co-consuls, Cambacérès and Lebrun, were quite willing to fulfill this circumscribed role. Lebrun was an aged conservative with some experience in finance while Cambacérès was a moderate revolutionary with a good deal of knowledge about the law. Both men thus covered two important bases for Bonaparte; they had enough of a political pedigree to be a sop to various French political factions, but they lacked the drive or ambition to seriously challenge the First Consul's authority. Lebrun, as the Third Consul, generally did not find much to do within the government beyond setting French finances in order. In contrast to Lebrun, Cambacérès proved to be an able servitor for Napoleon and was quite active in both the Consulate and Empire. As Second Consul, Cambacérès acted as an arch-chancellor for Naploleon (which indeed became his title under the Empire) and ruled France in his name while Napoleon was afield. The Second Consul monitored the legislature, which was a pliant body under the Constitution, conveyed the Bulletins to the press, and otherwise acted upon Napoleon's instructions. The Second Consul also appointed the subprefects and other bureaucrats and likewise filled in the gaps of Napoleon's decrees and edicts. The Second Consul's biggest contribution to Consulate policy was the authoring of the Napoleonic code and his legal acumen was useful for Napoleon when he sought to sidestep the strictures of French law. Cambacérès emerged in many respects a civilian equivalent of Berthier in that he translated his master's orders to the various prefects and other subordinates but had relatively little latitude to craft policy upon his own.

So while the Consuls were nominal executives of the Consulate, in reality they were elevated bureaucrats. As such, they possessed a degree of power within the French system, but such power was intertwined with the Napoleonic state. As both Second Consul and Arch-chancellor, Cambacérès was arguably the second most powerful man in France, but this power paled before that of Napoleon.

Sources

Dwyer, Philip G. Citizen Emperor: Napoleon in Power. New Haven: Yale University Press, 2013.

Woloch, Isser. Napoleon and His Collaborators: The Making of a Dictatorship. New York: W.W. Norton, 2001.