  We continue our study of the mapping ideal of operator $p$-compact maps,
previously introduced by the authors. Our approach embraces a more geometric
perspective, delving into the interplay between operator $p$-compact mappings
and matrix sets, specifically we provide a quantitative notion of operator
$p$-compactness for the latter. In particular, we consider operator
$p$-compactness in the bidual and its relation with this property in the
original space. Also, we deepen our understanding of the connections between
these mapping ideals and other significant ones (e.g., completely $p$-summing,
completely $p$-nuclear).
