Does $Y$ $\mathcal{F_t}$-adapted and $X$ adapted to $Y$ imply that $X$ is $\mathcal{F_t}$-adapted?
Solution 1:
Yes, $X$ is adapted to $(\mathcal{F}_t)_t$, because $\mathcal{F}^{Y}_t\subset \mathcal{F}_t$ necessarily for each $t$, otherwise $Y$ could not be adapted to $(\mathcal{F}_t)_t$.
(In general $(\mathcal{F}^Y_t)_t$ is known as the natural filtration of $Y$.)