# Commutative diagrams and arrows

I'm trying to make a commutative diagram with tikzcd. Here's what I get:

I want the hook from B to A to be upside down and the /simeq symbol over the arrow from C to D.

Sample code:

\documentclass[a4paper]{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\begin{document}
$\begin{tikzcd} A & B \arrow[l, hookrightarrow] \\ D \arrow[u, hookrightarrow] & C \arrow[l, "\simeq"] \arrow[u, hookrightarrow] \end{tikzcd}$
\end{document}


\documentclass[a4paper]{article}
$\begin{tikzcd} A \arrow[r, hookleftarrow] & B\\ D \arrow[u, hookrightarrow] & C \arrow[l, "\simeq", swap] \arrow[u, hookrightarrow] \end{tikzcd}$