In other words, given ordered sets $ A $ and $ B $ , and denoting by $ \mathscr{A} $ and $ \mathscr{B} $ the corresponding categories, show that a functor $ \mathscr{A} \to \mathscr{B} $ amounts to an order - preserving map $ A \to B $ . Two categories $ \mathscr{A} $ and $ \mathscr{B} $ are isomorphic, written as $ \mathscr{A} \cong \mathscr{B} $ , if they are isomorphic as objects of $ \mathbf{CAT} $ . Is there a functor $ Z {\colon}\linebreak[0] \mathbf{Grp} \to \mathbf{Grp} $ with the property that $ Z(G) $ is the centre of $ G $ for all groups $ G $ ?