If a fan moves the air and thus adds energy to the room, shouldn't the temperature go up, not down?
Of course the fan does the mechanical (rotational) work on the air thus increases the internal energy of air in the room which results in rise of temperature of the room. But one should know that moving air removes the heat energy from out skin by convection which makes one feel colder but actually the room temperature increases with the time as fan does work on the room air according to first law of thermodynamics
Assuming no heat loss from the room i.e. $Q=0$, $$0=\Delta +W$$ $$\Delta U=-W$$ The work done $W$ by the fan on the air is negative thus $\Delta U=-W$ will be positive i.e. the fan increases internal energy of the air molecules in the room which causes rise in temperature.