Consider the compact convex set $X = \{ x \in \mathbb{R}^n \mid x \geq 0, \ \underline{1}^\top x = 1 \}$.
I am wondering if the projection onto $X$ is the composition of the projection on $[0,1]^n$ and the projection onto $\{x \mid \underline{1}^\top x = 1\}$, namely if $$ P_X(\cdot) = P_{ [0,1]^n }\left( P_{ \{x \in \mathbb{R}^n \mid \underline{1}^\top x = 1\} }(\cdot) \right). $$
Comment: the projection onto $X$ is defined as $P_X(x) := \arg \min_{y \in X} \left\| x-y\right\| $.
No, it is not.
$H=\{x\;|\;\underline 1^\top x=1\}$ is a hyperplane, $[0,1]^n$ a hypercube, and $X$ is a simplex which results from intersecting these two. But the problem is that the second projection may well leave the hyperplane and therefore result in a point outside the simplex.
Here is an example for $n=3$: Consider the point $(8,8,-15)$ which already lies in $H$. Its projection onto $X$ would be $(\frac12,\frac12,0)$ which has distance $\frac{15}2\sqrt6\approx18.4$. But the corner of the cube at $(1,1,0)$ has distance $\sqrt{323}\approx18.0$ so it is closer.