Definition
Let be a category with subobject classifiers. Let be a morphism. The range of is given by such that the following diagram is a pullback:
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
a \arrow[d, "!"] \arrow[r, "f"] & b \arrow[d, "\phi"] \\
1 \arrow[r, "t"] & \Omega
\end{tikzcd}
\end{document}