For a map $f\colon X\to Y$ of quasi-compact quasi-separated schemes, we discuss
quasi-perfection, i.e., the right adjoint $f^\times$ of
$\mathbf Rf_*$ respects small direct sums.
This is equivalent to the existence of a functorial isomorphism
$f^\times\mathcal O_{Y}\otimes^{\mathbf L} \mathbf Lf^*(\<-\<)\!
{\longrightarrow{}^\sim} f^\times (-)$;
to quasi-properness
(preservation by $\Rf$ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of $\mathbf Lf^*\<$ (finite tor-dimensionality),
or of the functor $f^\times\<$; and to some other conditions. We use a globalization, previously known only for divisorial schemes, of
the local definition of pseudo-coherence of complexes,
as well as a refinement of the known fact that the derived
category of complexes with quasi-coherent homology is generated by a single
perfect complex.