Essential torus curves with the same primitive class are isotopic

Choose a smooth parametrization $h:T\to {\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$, as supplied by saying that $T$ is a smooth torus. It preserves essentiality, and its induced integral homology isomorphism preserves primitivity and equality up to sign. Apply [F1] to $h(c)$ to obtain primitivity of $[c]$. This is one supplied finite-dimensional parametrization, not a set-indexed choice.

If $[c]=\pm [d]$, apply [F1] to their images and conjugate its ambient isotopy by $h$. This gives the required ambient isotopy in $T$. If the original curves are disjoint and distinct, the two complementary annulus parameters in [F1] pull back by $h$; either supplies an embedded annulus with exactly those original curves as boundary. No annulus is inferred for intersecting original curves, and no choice axiom is introduced.