Don #167, #169 - the two definitions are equivalent. If the two partitions have the same cardinality then you can find an \\(h: P \rightarrow Q\\), but not necessarily satisfying \\(f.h=g\\).

For example, we can represent the partitions from the left and middle of the middle row by:

$$ f(\bullet)=f(\circ)=1, f(\ast)=2 $$

and

$$ g(\bullet)=g(\ast)=1, g(\circ)=2 $$

where \\(P=Q=\\{1, 2\\}\\).

Then there is no function \\(h: \\{1,2\\} \rightarrow \\{1,2\\}\\) such that \\(f.h=g\\) (since \\(f(\bullet)=f(\circ)\\), hence \\(h(f(\bullet))=h(f(\circ))\\), but \\(g(\bullet) \neq g(\circ)\\)).

For example, we can represent the partitions from the left and middle of the middle row by:

$$ f(\bullet)=f(\circ)=1, f(\ast)=2 $$

and

$$ g(\bullet)=g(\ast)=1, g(\circ)=2 $$

where \\(P=Q=\\{1, 2\\}\\).

Then there is no function \\(h: \\{1,2\\} \rightarrow \\{1,2\\}\\) such that \\(f.h=g\\) (since \\(f(\bullet)=f(\circ)\\), hence \\(h(f(\bullet))=h(f(\circ))\\), but \\(g(\bullet) \neq g(\circ)\\)).