Motivation to revisit the Conley index theory for discrete multivalued
dynamical systems stems from the needs of broader real applications, in
particular in sampled dynamics or in combinatorial dynamics. The new
construction of the index in [B. Batko and M. Mrozek, {\em SIAM J. Applied
Dynamical Systems}, 15(2016), pp. 1143-1162] based on weak index pairs, under
the circumstances of the absence of index pairs caused by relaxing the
isolation property, seems to be a promising step towards this direction. The
present paper is a direct continuation of [B. Batko and M. Mrozek, {\em SIAM J.
Applied Dynamical Systems}, 15(2016), pp. 1143-1162] and concerns properties of
the index defined therin, namely Wa\.zewski property, the additivity property,
the homotopy (continuation) property and the commutativity property. We also
present the construction of weak index pairs in an isolating block
