There is no wiggle room, a single doctrine is a single doctrine. Tac doctrine from a demi-company is different from a tac doctrine from a um CAD; this is specified with the different sources(if you were looking at the single doctrine as the type, then all tac doctrines would activate on the first declaration; which is counter to the rest of the rules and the example).
You can only activate the demi-company, um, or gladius at any given time, and it will only effect those units.
As another note: the um rule states all ums in the army benefit when "these doctrines" are activated; that means the doctrines provided by the um rule(so a primary detachment demi of if; with an allied um and a um formation woul have 2 sets of doctrines, 1 from demi and 1 from ct:um, but the ums would only benefit from the um set, not the demi set)
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