Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the homotopy-1-categorical version. In order to solicit an answer to that version, I repost my question here.
Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group spaces.
The inclusion $\mathrm{hoSp}_{\geq 0} \hookrightarrow \mathrm{hoSp}$ has a right adjoint, sending a spectrum $T$ to its connective cover $T\langle0\rangle$. In other words, for any connective spectrum $A$, we have $$ [A, T] = [A, T\langle 0\rangle].$$ I'm wondering about intermediate categories $\mathrm{hoSp}_{\geq 0} \hookrightarrow \mathcal{C} \hookrightarrow \mathrm{hoSp}$ for which the inclusion $\mathrm{hoSp}_{\geq 0} \hookrightarrow \mathcal{C}$ also has a left adjoint. Explicitly: for which spectra $T$ is there a connective spectrum $T'$ such that $$[T,A] = [T', A]$$ for all connective $A$? This $T'$ would be some sort of universal "quotient" of $T$ that quotients it down to being connective.
Dually, I can consider the full subcategory $\mathrm{hoSp}_{\leq 0} \hookrightarrow \mathrm{hoSp}$ of coconnective spectra. This inclusion has a left adjoint given by the truncation $T \mapsto \tau_{\leq 0}T$. On which intermediate category does this inclusion also have a right adjoint?
