  On a smooth line bundle $L$ over a compact K\"ahler Riemann surface $\Sigma$,
we study the family of vortex equations with a parameter $s$. For each $s \in
[1,\infty]$, we invoke techniques in \cite{Br} by turning the $s$-vortex
equation into an $s$-dependent elliptic partial differential equation, studied
in \cite{kw}, providing an explicit moduli space description of the space of
gauge classes of solutions. We are particularly interested in the bijective
correspondence between the open subset of vortices without common zeros and the
space of holomorphic maps. For each $s$, the correspondence is uniquely
determined by a smooth function $u_s$ on $\Sigma$, and we confirm its
convergent behaviors as $s \to \infty$. Our results prove a conjecture posed by
Baptista in \cite{Ba}, stating that the $s$-dependent correspondence is an
isometry between the open subsets when $s=\infty$, with $L^2$ metrics
appropriately defined.
