Pseudocovering and digital covering spaces

The notions of a local $(k_0,k_1)$-isomorphism and a weakly local $(k_0,k_1)$-isomorphism play crucial roles in developing a digital $(k_0,k_1)$-covering space and a pseudo-$(k_0,k_1)$-covering space, respectively. In relation to the study of pseudo-$(k_0,k_1)$-covering spaces, since there are some works to be refined and improved in the literature, the recent paper \cite{H10} improved and corrected some mistakes occurred in the literature. One of the important things is that the notion of a pseudo-$(k_0,k_1)$-covering map in \cite{H6,H9} was revised to be more broadened in \cite{H10}. Thus this new version is proved to be equivalent to a weakly local $(k_0,k_1)$-isomorphic surjection \cite{H10}. The present paper contains some works in \cite{H10} and we only deals with $k$-connected digital images $(X, k)$.