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?

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 (topological) direct limit of $M_{n}(\mathbb{C}),s$, with a coalgebraic structure?