Change description environment indents

Question

I want to set (now wrong) description environment indents the same as (already correct) enumerate environment indents.

Note that I already use \usepackage{enumitem}.

How?

\documentclass{article}
\usepackage{enumitem}

\begin{document}

\begin{enumerate}
  \item a.
  \item b.
\end{enumerate}

\begin{description}
  \item[1.] a.
  \item[2.] b.
\end{description}

\end{document}

The indents of two lists are different. I want to change description indent to become the same as enumerate indent.

ADDENDUM

I experimented with the below amsbook LaTeX file. In some reason, the indent of enumerate and description has not become the same as I desire. Please help me to find my error.

\documentclass{amsbook}
\usepackage{enumitem}

\setlist[enumerate,1]{leftmargin=1.7em,labelindent=*}
\setlist[description,1]{leftmargin=1.7em,labelindent=*}

\begin{document}

\begin{enumerate}
  \item a.
  \item b.
\end{enumerate}

\begin{description}
  \item[1.] a.
  \item[2.] b.
\end{description}

\end{document}

ADDENDUM 2

By @Harish Kumar recommendation, I've also tried the following. The result is not ideal, enumerate and description do not have the same left margin.

\documentclass{amsbook}
\usepackage{enumitem}

\renewcommand{\descriptionlabel}[1]{%
%   \hspace\labelsep \upshape #1.%
  \upshape #1.%
}
\setlist[enumerate,1]{leftmargin=2em,labelindent=*,label=\arabic*$^\circ$.,
ref=\arabic*$^\circ$} 
\setlist[description,1]{leftmargin=1em,labelindent=*}

\begin{document}

\begin{enumerate}
  \item \label{a} a.
  \item \label{b} b.
\end{enumerate}

\begin{description}
  \item[\ref{a}] a.
  \item[\ref{b}] b.
\end{description}

\end{document}

Answer 1

\documentclass{article}
\usepackage{enumitem,showframe}

\begin{document}

\begin{enumerate}
  \item a.
  \item b.
\end{enumerate}

\begin{description}[leftmargin=1.7em,labelindent=*,before={\renewcommand\makelabel[1]{##1}}]
  \item[1.] a.
  \item[2.] b.
\end{description}

\end{document}

Answer 2

The request being made is not at all clear: if one just wants to typeset the labels of a description environment as if they were part of an enumerate environment, why should one use description at all? Evidently, the question lacks some essential information. A possible interpretation is that what is actually wanted here is to typeset a regular description-style label with a number sticking in the margin, with an alignment similar to that of the labels of enumerate. The following example clarifies what I mean.

\documentclass[a4paper]{amsbook}
\usepackage[T1]{fontenc}
\usepackage{enumitem} % actually redundant, in this example

\newenvironment*{descriptionThatMimicksItemize}{%
  \list{}{\let\makelabel\descriptionTMIlabel}%
}{\endlist}
\newcommand*{\descriptionTMIlabel}[1]{\hss\llap{\upshape #1}}
\makeatletter
  \@ifdefinable\origLabelWidth{\newdimen\origLabelWidth}
  \@ifdefinable\origLabelSep  {\newdimen\origLabelSep}
\makeatother
\newcommand*{\myNumericLabel}[1]{%
  \makebox[\origLabelWidth][r]{#1}\hspace{\origLabelSep}%
}

\begin{document}

\origLabelWidth = \labelwidth
\origLabelSep   = \labelsep
% This is an over-simplification, but for now we can accept it.

The \texttt{enumerate} environment:
\begin{enumerate}
  \item \label{a}
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
  \item \label{b}
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
\end{enumerate}

The first solution:
\begin{descriptionThatMimicksItemize}
  \item[(\ref{a}) Is it this what you want?]
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
  \item[(\ref{b}) I wouldn't say so.]
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
\end{descriptionThatMimicksItemize}

The second one:
\begin{description}
  \item[\myNumericLabel{(\ref{a})}I'd rather say it's this]
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
  \item[\myNumericLabel{(\ref{b})}A label is needed, though]
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
\end{description}

\end{document}

Output:

If this is the real issue, it involves splitting the supplied label in two components, and therefore goes beyond the mere problem of setting the margins and the label width.

In conlusion, I think that it is impossible to fully answer this question until it is made clear what the precise request is.

Addition

I'm beginning to wonder whether it couldn’t be simply this what is actually wanted:

\documentclass[a4paper]{amsbook} % or whatever
\usepackage[T1]{fontenc}
\usepackage{hyperref}

\begin{document}

Let \( f\colon A\to B \) be a function.  The following are equivalent:
\begin{enumerate}
  \item\label{a}  $f$ is both injective and surjective;
  \item\label{b}  $f$ is invertible.
\end{enumerate}
Proof:
\begin{enumerate}
  \item[$\ref{a}\Rightarrow\ref{b}$:]
    Pick an element \( b\in B \), and let \( a\in A \) be the unique element 
    that is sent into~$b$ by~$f$\ldots
  \item[$\ref{b}\Rightarrow\ref{a}$:]
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
\end{enumerate}

\end{document}

Output:

Of course, the hyperlins work as expected.

But indeed, this wouldn’t be worth a bounty…

Second Addition

Yet another possibility:

\documentclass[a4paper]{amsbook} % or whatever
\usepackage[T1]{fontenc}
\usepackage{enumitem}
\usepackage{hyperref}

\setlist[enumerate,1]
  {align=left, labelindent=.5em, labelwidth=1.5em}

\begin{document}

Let \( f\colon A\to B \) be a function.  The following are equivalent:
\begin{enumerate}
  \item\label{a}  $f$ is both injective and surjective;
  \item\label{b}  $f$ is invertible.
\end{enumerate}
Proof:
\begin{enumerate}
  \item[$\mbox{(\ref{a})}\Rightarrow\mbox{(\ref{b})}$:]
    Pick an element \( b\in B \), and let \( a\in A \) be the unique element 
    that is sent into~$b$ by~$f$\ldots
  \item[$\mbox{(\ref{a})}\Rightarrow\mbox{(\ref{b})}$:]
    Lorem ipsum dolor sit amet consectetur adipisci elit iam 
    fugiaet enim duscula.  Reprivit ursula tum donec.
\end{enumerate}

\end{document}

And here is the output #3: