  We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and
comodules -- in an arbitrary monoidal DG category or bicategory by rewriting
their definitions in terms of unbounded twisted complexes. We develop new
notions of strong homotopy unitality and bimodule homotopy unitality to work in
this level of generality. For a strong homotopy unital $A_{\infty}$-algebra we
construct Free-Forgetful homotopy adjunction, its Kleisli category, and its
derived category of modules. Analogous constructions for
$A_{\infty}$-coalgebras require bicomodule homotopy counitality. We define
homotopy adjunction for $A_{\infty}$-algebra and $A_{\infty}$-coalgebra and
show such pair to be derived module-comodule equivalent. As an application, we
obtain the notions of an $A_{\infty}$-monad and of an enhanced exact monad. We
also show that for any adjoint triple $(L,F,R)$ of functors between enhanced
triangulated categories the adjunction monad $RF$ and the adjunction comonad
$LF$ are derived module-comodule equivalent.
