Very curved arrow with tikzcd

Question

Problem

I would like the arrow in the diagram

to extend farther to the right, so that it doesn't overlap with the long formula I have in the middle.

Current code

I am using the bend right option:

  \begin{equation*}
    \begin{tikzcd}
        A
        &
      \\
        BBBBBBBBB \ar{u}
        &
      \\
        C \ar{u}\ar[uu,"f^*",bend right=90,swap]
        &
    \end{tikzcd}
  \end{equation*}

Attempted solution

I attempted to use a control:

  \begin{equation*}
    \begin{tikzcd}
        A
        &
      \\
        BBBBBBBBB \ar{u}
        &
      \\
        C \ar{u}\ar[uu,"f^*",to path={..controls +(2.5,1.5) .. (\tikztotarget)},swap]
        &
    \end{tikzcd}
  \end{equation*}

This produces a diagram that looks like:

It's not overlapping anymore, but it doesn't look great, and more importantly the label $f^*$ disappeared.

Answer 1

above image exploit option looseness from TikZ:

\documentclass[border=3mm,
               preview
               ]{standalone}
\usepackage{tikz-cd}

\begin{document}
\[
\begin{tikzcd}
    A
    &
  \\
    BBBBBBBBB \ar{u}
    &
  \\
    C \ar{u}\ar[uu,"f^*",bend right=90,looseness=2,swap]
    &
\end{tikzcd}
\]
\end{document}