Unit holds because \\(0 + m = m \\) and \\(m + 0 = m \\),
associativity holds because \\((m_1 + m_2) + m_3 = m_1 + (m_2 + m_3) \\)
Unit holds because \\( ++ m = m \\) and \\( m ++  = m \\)
associativity holds because \\((m_1 ++ m_2) ++ m_3 = m_1 ++ (m_2 ++ m_3) \\)
A monoid can be viewed as a category with a single object (call it obj)
with the set of morphisms C(obj, obj) = M
the function * gives us the composition rule
and e in M gives us the identity morphism.
The Unit and associative laws of the Moniod correspond to the unit and associative laws of the category.