The hybrid Phillips curve type equation

so I’m trying to replicate Trigari(2009) (Equilbrium unemployment, job flows and inflation dynamics) and the log linear system of equations given in the paper features a hybrid phillips curve type equation with a backward looking element ala Gali Gertler (1999). I’m trying to derive it but I get stuck. Does anynone know of literature where the derivation is more explicit?
In addition, I cannot get my head around the fact that habit formation in consumption does not change the form of the phillips curve. The stochastic discount factor(beta*u’_c(t+1)/u’_c(t)), which must be plugged in eventually, becomes a lot less trivial I think.
If this is not the right forum for such theoretical questions then please give me a tip as to where I could ask such things.
Much obliged

The standard reference for that type of exercise would be Gali’s textbook. However, his derivations are often also only coarse. I am attaching my lecture slides that should fill his gaps.

The reason that habits do not matter is that during linearization the part related to the stochastic discount factor drops out, because it is multiplied by something that is 0.
Derivation New Keynesian Phillips Curve.pdf (527 KB)

Thank you!