I'm reading Waldhausen's 1985 Algebraic K-Theory of Spaces and have not been able to follow his argument for Corollary 2 of Lemma 1.4.1. In Waldhausen's notation. $s_\bullet\mathcal{C}$ is the simplicial set where $s_n \mathcal{C}$ is the set of objects of $S_n\mathcal{C}$. Lemma 1.4.1 says "An exact functor of categories with cofibrations $f\colon \mathcal{C} \rightarrow \mathcal{C}'$ induces a map $s_\bullet f \colon s_\bullet\mathcal{C} \rightarrow s_\bullet\mathcal{C}'$. An isomorphism between two such functors $f$ and $f'$ induces a homotopy between $s_\bullet f$ and $s_\bullet f'$". Corollary 1 says that an exact equivalence of categories with cofibrations $\mathcal{C} \rightarrow \mathcal{C}'$ induces a homotopy equivalence $s_\bullet \mathcal{C} \rightarrow s_\bullet \mathcal{C}'$. Corollary 2 say that if $i$ are the isomorphisms of $\mathcal{C}$, then there is a homotopy equivalence $s_\bullet \mathcal{C} \rightarrow iS_\bullet \mathcal{C}$. To justify 2 Waldhausen says consider "the simplicial object $[m] \mapsto i_mS_\bullet\mathcal{C}$, the nerve of $iS_\bullet\mathcal{C}$, in the $i$-direction" and note "that $i_0S_\bullet \mathcal{C} = s_\bullet \mathcal{C}$ and that the face and degeneracy maps are homotopy equivalences by" Corollary 1.
It seems to me that nerve in the $i$-direction means that $i_mS_\bullet\mathcal{C}$ is the simplicial set $\text{Fun}([m], iS_\bullet\mathcal{C})$.I think face and degeneracy maps refers to those of this simplicial set. But how does Corollary 1 imply that these are homotopy equivalences and how does them being homotopy equivalences give the result?