Assume I have a stochastic process $X(t)$ adapted to its natural filtration and define the value function
$$ F(t,x) = \mathbb E[ g(X(T))\: |\: X(t) = x ] $$
For some $T>0$. Moreover, define the function
$$ G(s,y) = \mathbb E[ h(F(t, X(t))) \: |\: X(s) = y ], $$
for $T \geq t \geq s$. Then, is this true?
$$ G(s,y) = h(F(s,y)). $$
I believe so, but I was not able to show it.