= Solution
For two candidates $A,C$, their difference is a continuous <adapted> <finite-variation process>. It is also a <local martingale>, since
$$
A-C=(Y^2-C)-(Y^2-A).
$$
Apply part (b) after subtracting its initial value. Consequently
$$
\boxed{A_t-C_t=A_0-C_0\quad\text{for all }t\text{ almost surely}.}
$$
This proves the <uniqueness of an increasing square compensator> when the initial value is prescribed, in particular under the standard <quadratic variation> normalization $A_0=C_0=0$.
The printed assertion omits that normalization and is literally false. For standard <Brownian motion> $Y$, both $A_t=t$ and $C_t=t+1$ are continuous <adapted> increasing processes; both $Y_t^2-t$ and $Y_t^2-t-1$ are <martingales>. Thus \b[uniqueness holds after fixing the initial value; otherwise it holds only up to an initial constant].
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