  Let $(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points
$\mathbb{M}\subseteq \partial\Sigma\neq\varnothing$ and punctures
$\mathbb{P}\subseteq\Sigma\setminus\partial\Sigma$. In this paper we show that
for every curve $\gamma$ on $\Sigma\setminus\mathbb{P}$, the curve obtained by
resolving the kinks of $\gamma$ in any order is uniquely determined, up to
homotopy in $\Sigma\setminus\mathbb{P}$, by the $2$-orbifold homotopy class of
$\gamma$, in which the punctures are interpreted to be orbifold points of order
$2$. Our proof resorts to an application of the Diamond Lemma.
