  Dvo\v{r}\'{a}k and Postle \cite{DP} introduced a \textit{DP-coloring} of a
simple graph as a generalization of a list-coloring. They proved a Brooks' type
theorem for a DP-coloring, and Bernshteyn, Kostochka and Pron \cite{BKP}
extended it to a DP-coloring of multigraphs. However, detailed structure when a
multigraph does not admit a DP-coloring was not specified in \cite{BKP}. In
this note, we make this point clear and give the complete structure. This is
also motivated by the relation to signed coloring of signed graphs.
