Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$???

Given $f,g \in C$ and $1\leq k \leq m$ can we have
```
\begin{align*}\Delta^{m-1}(fg)&=\Delta^{k-1}(fg) \otimes id^{m-k} + \Delta^{k-1}(f)\otimes \Delta^{m-k-1}(g) \\
&+\Delta^{k-1}(g)\otimes \Delta^{m-k-1}(f)+id^{\otimes k} \otimes \Delta^{m-k-1}(fg)???\end{align*}
```

Thanks,