Long equals in tikz

You can use double equal sign distance in your \draw options to set the distance between the double lines to match that of an equal sign.

The double lines work with any path specification, including things like bend left or to [out=45]:

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw [double equal sign distance] (0,0) to [out=45, in=180] (3,0) node [anchor=mid west] {$= A$};
\end{tikzpicture}
\end{document}

The great tikz-cd package (tikz-cd, manual) includes the option equal for arrows.

Code

\documentclass[tikz,convert=false]{standalone}
\usepackage{tikz-cd}
\usetikzlibrary{arrows}
\tikzset{commutative diagrams/.cd,arrow style=tikz,diagrams={>=latex'}}
\begin{document}
\begin{tikzcd}[swap,bend angle=45]
A \dar{f} \rar{a} \ar[bend left,equal]{rr}
  & X \dar{g} \rar{r}
      & A \dar[swap]{f} \\
B \rar[swap]{i}   \ar[bend right,equal]{rr}
      & Y \rar[swap]{\beta}
          & B
\end{tikzcd}
\end{document}

Output