Aligning multiple function definitions

Question

I wish to define two functions, versions of addition and multiplication in an object similar to a ring of fractions. Is it better to align on the colons, the arrows, or define each function separately? Also, should the display of the domains and codomains be aligned with the value assignments? I am using poor man's bold to distinguish these operations from operations the object that corresponds to the original ring. Here is a mwe.

    \documentclass[paper=letter,fontsize=12pt]{scrreprt}
    \usepackage[utf8]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsthm}
    \usepackage{amssymb}
    \usepackage{mathtools}
    \begin{document}
    Define the functions 
    \begin{align*}
    &\pmb{+} \colon \mathsf{BF} \times \mathsf{BF} \longrightarrow \mathsf{BF} \text{ and} \\
    &\pmb{\circ} \colon \mathsf{BF} \times \mathsf{BF} \longmapsto \mathsf{BF}
    \end{align*}
    for each $\big( (a, u), (b, v) \big) \in \mathsf{BF} \times \mathsf{BF}$ by the assignments
    \begin{align}
    &\pmb{+} \colon \big( (a, u)(b, v) \big) \longmapsto \big( (a \cdot v) + (u \cdot v), u \cdot v \big) \\
    &\pmb{\circ} \colon \big( (a, u)(b, v) \big) \longmapsto \big( a \cdot b, u \cdot v \big)
    \end{align}
    \end{document}

Answer 1

Like this?

\documentclass[paper=letter,fontsize=12pt]{scrreprt}
\usepackage{mathtools}
\usepackage{amssymb, amsthm}

\begin{document}
Define the functions
    \begin{align}
\pmb{+}     & \colon \mathsf{BF} \times \mathsf{BF} 
                \longrightarrow \mathsf{BF} \text{ and}                     \notag\\
\pmb{\circ} & \colon \mathsf{BF} \times \mathsf{BF} \longmapsto \mathsf{BF} \notag
%
\intertext{for each $\big((a,u),(b,v)\big)\in\mathsf{BF}\times\mathsf{BF}$ by the assignments}
%
\pmb{+}     & \colon \big((a,u)(b,v)\big)\longmapsto \big((a\cdot v)+(u\cdot v), u\cdot v\big) \\
\pmb{\circ} & \colon \big((a,u)(b,v)\big)\longmapsto \big( a\cdot b, u\cdot v\big)
    \end{align}
\end{document}