Getting extremely large arrows with tikzcd

Add a width specification:

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

\DeclareMathOperator{\ob}{ob}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\cod}{cod}

\begin{document}

\begin{tikzcd}
E \arrow[r, "e"] &
  \prod_{i \in \ob(\mathcal{E})} F(i)
  \arrow[
    r, shift left,
    "{\langle\pi_{\cod(u)} \mid u \in \hom(\mathcal{E})\rangle}"
  ]
  \arrow[
    r, shift right,
    "{\langle F(u)\pi_{\dom(u)} \mid u \in \hom(\mathcal{E})\rangle}"'
  ] &[8em]
  \prod_{u \in \hom(\mathcal{E})} F(\cod(u))
\end{tikzcd}

\end{document}

Note the adjustments I made:

• <...> should be \langle...\rangle
• \Pi should be \prod
• \ | \ should be \mid
• ob, dom and cod should be typeset upright

A possible alternative with \smashoperator that makes the diagram shorter:

\documentclass{article}
\usepackage{amsmath,mathtools}
\usepackage{amssymb}
\usepackage{tikz-cd}

\DeclareMathOperator{\ob}{ob}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\cod}{cod}

\begin{document}

\begin{tikzcd}
E \arrow[r, "e"] &
  \smashoperator{\prod_{i \in \ob(\mathcal{E})}} F(i)
  \arrow[
    r, shift left,
    "{\langle\pi_{\cod(u)} \mid u \in \hom(\mathcal{E})\rangle}"
  ]
  \arrow[
    r, shift right,
    "{\langle F(u)\pi_{\dom(u)} \mid u \in \hom(\mathcal{E})\rangle}"'
  ] &[10em]
  \smashoperator{\prod_{u \in \hom(\mathcal{E})}} F(\cod(u))
\end{tikzcd}

\end{document}

Here are two different ways

\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz-cd}

\DeclareMathOperator{\cod}{cod}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\ob}{ob}

\begin{document}

Option 1: wide column separation specified by dimension
\begin{equation*}
  \begin{tikzcd}[column sep = 10em]
    E \arrow[r, "e"] & \displaystyle\prod_{i \in
    \ob(\mathcal{E})} \mkern-12mu F(i)
    \arrow[r, shift left, " {\langle\pi_{\cod(u)} \mid u \in
    \hom(\mathcal{E})\rangle}"] \arrow[r, shift right, "{\langle F(u)
    \pi_{\dom(u)} \mid u \in
    \hom(\mathcal{E})\rangle}"']
    & {\displaystyle\prod_{u \in \hom(\mathcal{E})} \mkern-12mu F(\cod(u))}
  \end{tikzcd}
\end{equation*}

Option 2: let arrows cover two (or more) columns
\begin{equation*}
  \begin{tikzcd}[column sep = huge]
    E \arrow[r, "e"]
    & \displaystyle\prod_{i \in \ob(\mathcal{E})} \mkern-12mu F(i)
    \arrow[rr, shift left, " {\langle\pi_{\cod(u)} \mid u \in
    \hom(\mathcal{E})\rangle}"]
    \arrow[rr, shift right, "{\langle F(u) \pi_{\dom(u)}
    \mid u \in \hom(\mathcal{E})\rangle}"']
    && {\displaystyle\prod_{u \in \hom(\mathcal{E})} \mkern-12mu F(\cod(u))}
  \end{tikzcd}
\end{equation*}

\end{document}

The first is simply to specify the column sep as a dimension. The second is to add an extra column under the long arrows, so you have

A \arrow[r] & B \arrow[rr] && C

instead of

A \arrow[r] & B \arrow[r] & C

You can make this wider by using more columns

A \arrow[r] & B \arrow[rrr] &&& C

etc.

Note I have recoded much of your mathematics.

The above solutions corresponding to working with a fixed grid, which is often a good idea for consistent spacing. An alternative is the extra space syntax of egreg's answer:

&[5em]

in the first row, will push the subsequent column 5em to the right and so making the corresponding arrows longer.

Add an empty column between columns 2 and 3. I also added some improvements to the diagram.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usepackage{fullpage}
\usepackage{amsthm}
\usepackage{adjustbox}

\begin{document}

\begin{tikzcd}[column sep= huge, row sep=huge]
E \arrow[r, "e"] & \Pi_{i \in ob(\mathcal{E})} F(i) \arrow[rr, shift left=1ex, "
{\langle\pi_{cod(u)} \, | \, u \in \hom(\mathcal{E})\rangle}"] \arrow[rr, shift right=1ex, "{\langle F(u)
\pi_{dom(u)} \, | \, u \in \hom(\mathcal{E})\rangle}"'] & & {\Pi_{u \in \hom(\mathcal{E})
F(cod(u))}}
\end{tikzcd}

\end{document}