  A classification theorem for three different sorts of Mal'tsev categories is
proven. The theorem provides a classification for Mal'tsev category, naturally
Malt'sev category, and weakly Mal'tsev category in terms of classifying classes
of spans. The class of all spans characterizes naturally Mal'tsev categories.
The class of relations (i.e. jointly monomorphic spans) characterizes Mal'tsev
categories. The class of strong relations (i.e. jointly strongly monomorphic
spans) characterizes weakly Mal'tsev categories. The result is based on the
uniqueness of internal categorical structures such as internal category and
internal groupoid (Lawvere condition). The uniqueness of these structures is
viewed as a property on their underlying reflexive graphs, restricted to the
classifying spans. The class of classifying spans is combined, via a new
compatibility condition, with split squares. This is analogous to orthogonality
between spans and cospans. The result is a general classifying scheme which
covers the main characterizations for Mal'tsev like categories. The class of
positive relations has recently been shown to characterize Goursat categories
and hence it is a new example that fits in this general scheme.
