Remark 1

When $C'$ is reduced to $C'_0=M$, then $(C\otimes_A C')_n=C_n\otimes_A M$ and $D=d\otimes 1_M$; for example $C\otimes_A A_s$ is naturally identified with $C$. Similarly, when $C$ is reduced to $C_0=P$, then $(C\otimes_A C')_n=P\otimes_A C'_n$ and $D=1_P\otimes d$.