EDIT: First, sorry about not checking nLab, I forget about that site far too often. Second, I should say that I have a little bit of motivation for my property two. So let me clarify what I meant in property two. Given a multiset, it can be thought of as a pair $S\times\mathbb{N}$ for a set $S$. Now, when considering morphisms between multisets I want the maps $f,g:\lbrace 1122\rbrace\rightarrow\lbrace34\rbrace$ such that $f$ sends
$\begin{eqnarray*}1&\mapsto& 3,\1&\mapsto& 4,\2&\mapsto& 3,\2&\mapsto& 4\\end{eqnarray*}$
and $g$ sends
$\begin{eqnarray*}1&\mapsto& 4,\1&\mapsto& 3,\2&\mapsto& 4,\2&\mapsto& 3\\end{eqnarray*}$
to be the same morphism. But if $h$ sends
$\begin{eqnarray*}1&\mapsto& 4,\1&\mapsto& 4,\2&\mapsto& 4,\2&\mapsto& 3\\end{eqnarray*}$
then $h$ is not the same as $g$ or $f$. Further I would like it such that $\lbrace 112\rbrace$ is not a subobject of $\lbrace 12\rbrace$ but it is a subobject of $\lbrace 11122\rbrace$.
Hopefully this will clear it up.
