Let $X$ be an $n$-connective spectrum for some $n\in \mathbb{Z}$. Is then $[X, Y] = [X, Y\langle n\rangle]$ for all spectra $Y$, where $Y\langle n\rangle$ denotes the $n$-connective cover of $Y$?

Dually, if $Y$ is $n$-truncated, i.e. $\pi_k(Y) = 0$ for all $k > n$, is then $[X, Y] = [X_{\leq n}, Y]$ for all spectra $X$, where $X_{\leq n}$ denotes the truncation of $Y$?

In other words, are taking the connected cover respectively truncation a right respectively left adjoint?

What about the mapping spectra?

\edit: I mixed up the second part as noted in the comments.