Reformat the mathematical product sign

Question

I'm pretty sure there is a simple formatting solution to this, I just can't find it anywhere.
I'll describe what I want to achieve with the code and the picture.

In my last post I was told to add code to my posts, to make it easier for others to help, so there it the relevant part:

\underline{Definition}: A \textbf{state} (or equivalently, an \textbf{instance})  of a database,
is a\\function $f:\Phi\rightarrow \prod_{D\in \mathbb{D}}D$.
Intuitively, $f$ takes a relation name $\phi\in \Phi$ and gives
it an interpretation as a table.

Which produces this:

Notice how the symbol D\in\mathbb{D} does not sit right under the \prod symbol as I intend.
It does sit there when I wrap it with double $$, but it also places the mathematical expression and basically ruins everything...:

and that's not something I want...

Is there a way to do it?

Answer 1

You can use the nccmath package and its \medop command (medium-sized operator, ~ 80 % of display style) You can have the index under the product symbol if you insert a new paragraph afterwards. Here are two ways of doing things. I use an unnumbered definition environment with the label in small caps, with the ntheorem package. Underlining titles dates back to the times of typewriters…

\documentclass[a4paper]{article}
\usepackage{amsmath, amsfonts}
\usepackage{nccmath}

\setlength\parindent{1em}
\usepackage[thmmarks, amsmath, thref]{ntheorem}
\theoremstyle{nonumberplain}
\theoremheaderfont{\scshape}
\theorembodyfont{\normalfont}
\theoremseparator{:}
\newtheorem{Def}{Definition}
\begin{document}

\begin{Def}
 A \textbf{state} (or equivalently, an \textbf{instance}) of a database, is a function $f \colon \Phi \longrightarrow \medop\prod  \limits _{D \in \mathbb{D}}D$.

Intuitively, $f$ takes a relation name $\phi \in \Phi$ and gives it an interpretation as a table.\end{Def}

\begin{Def}
 A \textbf{state} (or equivalently, an \textbf{instance}) of a database,is a function $f \colon \Phi \longrightarrow \medop\prod  _{D \in \mathbb{D}}D$. Intuitively, $f$ takes a relation name $\phi \in \Phi$ and gives it an interpretation as a table.\end{Def}
\end{document} 

Answer 2

I think that if you really want to (or need to) to place the limit term (D\in\mathbb{D}}) below the product symbol and thereby give it a lot of visual prominence, you should definitely go "all the way" and place the entire math expression on a line by itself, in display-math mode.

\documentclass{article}
\usepackage{amssymb} % for \mathbb macro
\setlength\textwidth{8cm} % just for this example
\begin{document}
\noindent
\underline{Definition}: A \textbf{state} (or equivalently, an \textbf{instance})  of a database is a function
\[f\colon\Phi\rightarrow \prod_{D\in \mathbb{D}}D\,.\]
Intuitively, $f$ takes a relation name $\phi\in \Phi$ and gives it an interpretation as a table.
\end{document}