$$ \begin{aligned}\mathbb{E}_{W|Z}\left[\mathbb{E}_{WY|W,Z}\left[WY\;|\;W,Z=z\right]\;|\;Z=z\right] = \mathbb{E}_{W|Z}\left[W\mathbb{E}_{Y|W, Z}\left[Y\;|\;W, Z=z\right]\;|\;Z=z\right] \end{aligned} $$
I'm wondering if this is a true statement where $W, Y, Z$ are random variables. But I'm not sure why exactly this is true (or false). My intuition is that $W$ becomes a constant but why does it become a constant?
The equality is true even without the outer conditional expectation. Loosely speaking, since $W$ is being conditioned on in the conditional expectation $E[\cdot \mid W, Z=z]$, it is "known" and thus can be treated as a non-random quantity: $E[WY \mid W, Z=z] = W E[Y \mid Z=z]$ is just the same as $E[2Y] = 2 E[Y]$. This can be made rigorous..