How to typset the following correpondence?

I am learning LaTeX. How can I typeset this correspondence?

\documentclass{article}
$\begin{split} \left[\parbox{4cm}{\raggedright \emph{Objects:} smooth curves defined over K\\ \emph{Maps:} nonconstant \dots}\right] & \rightsquigarrow \left[\parbox{4cm}{\raggedright \emph{Objects:} smooth curves defined over K\\ \emph{Maps:} nonconstant \dots}\right]\\ C/K & \rightsquigarrow K(C)\\ \phi: C_1\rightarrow C_2 & \rightsquigarrow \phi^\star: K(C_2)\rightarrow K(C_1) \end{split}$