Let $X$ be a $k$-connected spectrum.

I want to deduce how connected the counit of $(\Sigma^\infty, \Omega^\infty)$- adjunction is, that is, how connected is the map $$ \Sigma^\infty\Omega^\infty X \to X. $$

Any help would be appreciated.

Let $X$ be a $k$-connected spectrum for $k \in \Bbb{Z}$.

I want to deduce how connected the counit of $(\Sigma^\infty, \Omega^\infty)$- adjunction is, that is, how connected is the map $$ \Sigma^\infty\Omega^\infty X \to X. $$

Any help would be appreciated.