  We explore the interplay between t-structures in the bounded derived category
of finitely presented modules and the unbounded derived category of all modules
over a coherent ring $A$ using homotopy colimits. More precisely, we show that
every intermediate t-structure in $D^b(\operatorname{mod}(A))$ can be lifted to
a compactly generated t-structure in $D(\operatorname{Mod}(A))$, by closing the
aisle and the coaisle of the t-structure under directed homotopy colimits.
Conversely, we provide necessary and sufficient conditions for a compactly
generated t-structure in $D(\operatorname{Mod}(A))$ to restrict to an
intermediate t-structure in $D^b(\operatorname{mod}(A))$, thus describing which
t-structures can be obtained via lifting. We apply our results to the special
case of HRS-t-structures. Finally, we discuss various applications to silting
theory in the context of finite dimensional algebras.
