# Interaction Term in Fuzzy RD

I'm hoping someone can help me understand the intuition behind the interaction term in a fuzzy RD model. The setup is as follows:

$x$ = rating variable with discontinuity at $x = k$

$D$ = dummy=1 if $x > k$

$T$ = treatment variable; the probability of treatment is higher for $x > k$

$y$ = outcome variable

Following the standard procedure of treating a fuzzy RDD as local IV model (see for example Fuzzy RDD issue, also implemented by the R/Stata rdrobust package), I would set up the following equations:

First stage: $T = \beta_0 + \beta_1D + \beta_2x + \beta_3D*x + \epsilon$

Second stage: $y = \beta_0 + \beta_1\hat{T} + \beta_2x + \beta_3D*x + \epsilon$

I'm confused as to why $D$ ends up in the second stage equation, given that $D$ served as our instrument and generally it would violate the exclusion restriction to have an instrument in the second stage. I understand that we generally need the interaction term in RD models to allow for a different relationship between x and y on either side of the cutoff, but I do not understand how we can "think of fuzzy RDD as a local IV model" when the "instrument" appears in the second stage.