Overset alignment above text and inserted whitespace

Question

I'd like to insert an annotation above equations without breaking the spacing of that equation. The annotation should be aligned to the left, and begin with an arrow that points down on the first symbol.

Code

\newcommand{\hookdownleft}{\rotatebox[origin=b]{90}{$\Lsh$}}
\begin{dfn}
$\overset{\hookdownleft \text{family of sets from } \Omega \: 
\rightarrow \: \text{one set from } \Omega}{op} \in \mathcal{O} = 
\set{\text{set operations}}}$
\end{dfn}

Output

Desired

Answer 1

As pointed by Barbara Beeton, the simplest solution uses \mathrlapfrom mathtools inside the first argument of \overset. I added a \mathstrut to the second argument, to ensure a correct vertical spacing between the overset thing and the main line of equation:

\documentclass{article}

\usepackage{mathtools, amsthm, amssymb}
\DeclarePairedDelimiter{\set}\{\}
\usepackage{rotating, bigstrut}
    \newcommand{\hookdownleft}{\rotatebox[origin=b]{90}{$\Lsh$}}
\theoremstyle{definition}
\newtheorem{dfn}{Definition}

\begin{document}

\begin{dfn}
$\overset{\mathrlap{\hookdownleft \text{family of sets from } \Omega \:
\rightarrow \: \text{one set from } \Omega}}{op\mathstrut} \in \mathcal{O} =
\set{\text{set operations}}$
\end{dfn}

\end{document}