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)\$$).