Let $\left\{A_i\right\}_{i\in I}$ be a countable set of $C^*$-algebras $A_i$ and $ \bigoplus_{i\in\mathbb{N}}A_i$ the direct sum as a $C^*$-algebra. Let $A_i^1$ the unitization of $A_i$ and $c_0$ the $C^*$-algebra of zero-limit sequences.
I read that there is a split exact sequence
$$0\to \bigoplus_{i\in\mathbb{N}}A_i \to \bigoplus_{i\in\mathbb{N}}A_i^1\to c_0\to 0 $$ and my questions is:
Is it correct that the morphisms of the sequence above are:
The inclusion $ \bigoplus_{i\in\mathbb{N}}A_i \to \bigoplus_{i\in\mathbb{N}}A_i^1 ,\; (a_n)_n\mapsto \left(a_n+\lambda_n 1_{A_i^1}\right)_n$,
the projection $\bigoplus_{i\in\mathbb{N}}A_i^1\to c_0 \; \left(a_n+\lambda_n 1_{A_i^1}\right)_n\mapsto \left(\lambda_n 1_{A_i^1}\right)_n$ and that
$c_0 \to \bigoplus_{i\in\mathbb{N}}A_i^1 ,\; (z_n)_n\mapsto \left(z_n 1_{A_i^1}\right)_n$?
Or is something wrong?
The only thing that is wrong is that the inclusion $\bigoplus_{i\in\mathbb{N}}A_i \to \bigoplus_{i\in\mathbb{N}}A_i^1 $ is given by $$(a_n)_n\mapsto \left(a_n\right)_n.$$