= Solution
The transformation condition is necessary by the definition of a <cusp form>. For sufficiency, fix $\sigma\in SL_2(\mathbb Z)$ representing any <modular cusp>, and set $g=f|_k\sigma$. By item (i), $g$ is <holomorphic> in the upper half-plane. Choose an actual period $h$ with $\sigma T^h\sigma^{-1}=\gamma\in\Gamma$. Associativity of slash operations and the given transformation law give
$$
g|_kT^h=f|_k\sigma T^h=f|_k\gamma\sigma=(f|_k\gamma)|_k\sigma=g.
$$
Thus $g(\tau+h)=g(\tau)$ and $g$ is a single-valued <holomorphic function> on a punctured disk with coordinate $q_h=e^{2\pi i\tau/h}$. The calculation in part (b) transports the original uniform bound to this translate:
$$
|g(\tau)|y^{k/2}=\phi(\sigma\tau)\leq B.
$$
This puts every <modular cusp> in exactly the situation needed in item (iii). No assumption that a single <modular cusp> represents all of $\Gamma$'s <modular cusps> is made.
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