Monetary Transmission in New Keynesian - Euler Equation I get that NK model basically consists of the three equations: Taylor rule, Dynamic IS curve and the NK Phillips curve. 

I want to know the effect of nominal interest rate reduction on the transmission channels.

My lecturer says ''nominal rate goes down, real rate therefore rises, raising actual output relative to potential through the euler equation.  this then feeds into the philips curve and raises actual inflation.''

Q: How does the real rate rise when nominal rate goes down? And what is the Euler Equation here?
Real interest rate = Nominal interest rate - Expected inflation


Reducing the nominal interest rate by more than the decline in the expected inflation, with respect to the inflation target, leads the real interest rate to go down.
When the real interest rate goes down, the real exchange rate depreciates, both lead real monetary conditions to loosen. [Real monetary conditions = a1*RR_gap + (1*a1)*RER_gap].
Looser real monetary conditions lead to higher aggregate demand 'output gap' (higher domestic demand due to lower real interest rate and higher external demand due to the real exchange rate depreciation). [IS curve - Euler equation]
Higher aggregate demand puts more pressure on the real marginal costs, which in addition to higher inflation expectations that are directly affected by reducing the nominal interest rate, increase the inflation rate.