>Is there a  coalgebra  structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is  compatible  with the  natural  embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$  with    $i_{n}(A)= A\oplus 0$. That is $(i_{n}\otimes i_{n})\circ \Delta_{n}=\Delta_{n+1}\circ i_{n}$.


 If the answer is yes, can this  method be used to equip the  space  of  compact  operators, the  direct limit of  $M_{n}(\mathbb{C}),s$,   with  a coalgebraic  structure?