Commutative diagram with curved arrow

Question

I want to get a bent arrow from node A to node E, this is my code:

\usepackage{tikz-cd}
\usetikzlibrary{cd}
\usepackage{tikz}
\usetikzlibrary{positioning}

\begin{document}

\begin{center}
\begin{tikzpicture}

\node (E) at (0,0) {$T_V U$};
\node[right=of E] (F) {$U\times V$};
\node[below=of F] (A) {$U$};
\node [right=of F] (D) {};
\node [below=of D] (G) {$V$};


\draw[->] (E)--(F) node [midway,above] {$\simeq$};
\draw[->] (E)--(A) node [midway,below] {$f$};
\draw[->] (F)--(A) node [midway,right] {$\pi^2$};
\draw[->] (F)--(G) node [midway,right] {$\pi^1$};
\draw[->,bend] (A)--(E) node [midway,left] {$v$}


\end{tikzpicture}
\end{center}

\end{document}

Answer 1

I suggest using tikz-cd directly:

\documentclass{article}
\usepackage{tikz-cd}

\begin{document}

\[
\begin{tikzcd}
T_V U \arrow [dr,"f"] \arrow[r,"\simeq"] &
  U\times V \arrow[d,"\pi^2"] \arrow[dr,"\pi^1"] \\
& U \arrow[ul,bend left,"v"] & V
\end{tikzcd}
\]

\end{document}

Answer 2

For the sake of it, here is what went wrong with tikz version.

When you want to bend, you have to say bend left or bend right. and this operation needs either to or edge path, not --. The last point is that you have to use proper placements for nodes like abobe, above right etc as appropriate. When you use node soon after to, the default position will be midway hence you don't need to specify it. Altogether, this works:

\draw[->,bend left] (A) to node [left] {$v$} (E) ;

Full code:

\documentclass{article}
%\usepackage{tikz-cd}
%\usetikzlibrary{cd}
\usepackage{tikz}
\usetikzlibrary{positioning}

\begin{document}

\begin{center}
\begin{tikzpicture}

\node (E) at (0,0) {$T_V U$};
\node[right=of E] (F) {$U\times V$};
\node[below=of F] (A) {$U$};
\node [right=of F] (D) {};
\node [below=of D] (G) {$V$};


\draw[->] (E)--(F) node [midway,above] {$\simeq$};
\draw[->] (E)--(A) node [midway,above] {$f$};
\draw[->] (F)--(A) node [midway,right] {$\pi^2$};
\draw[->] (F)--(G) node [midway,above right] {$\pi^1$};
\draw[->,bend left] (A) to node [left] {$v$} (E) ;


\end{tikzpicture}
\end{center}

\end{document}

But for commutative diagrams, use tikz-cd and live happier.