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

$A_{\infty}$-structures in monoidal DG categories and strong homotopy unitality