Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there exist a $1$-Lipschitz map $\varphi$ from the interior of $(abc)$ to the interior of $(\overline{abc})$.

Using Kirszbraun theorem (for Alexandrov spaces, as stated by Lang and Schroeder or Alexander, Kapovitch and Petrunin) this is simple. Just identify the sides of $(abc)$ and $(\overline{abc})$ in the usual way, this map is $1$-Lipschitz because of the curvature bound and we can apply Kirszbraun theorem to extend it to the interior of the triangle.

I was wondering if one could explicitly build $\varphi$ in this specific case.