  For a finite dimensional algebra $A$, we prove that the bounded homotopy
category of projective $A$-modules and the bounded derived category of
$A$-modules are dual to each other via certain categories of locally-finite
cohomological functors. The duality gives rise to a $2$-categorical duality
between certain strict $2$-categories involving the bounded homotopy categories
and bounded derived categories, respectively. We apply the $2$-categorical
duality to the study of triangle autoequivalence groups. These results are
analogous to the ones in [M.R. Ballard, {\em Derived categories of sheaves on
singular schemes with an application to reconstruction}, Adv. Math. {\bf 227}
(2011), 895--919].
