  Let $V$ be a vertex operator algebra and $A^{\infty}(V)$ and $A^{N}(V)$ for
$N\in \mathbb{N}$ the associative algebras introduced by the author in [H5].
For a lower-bounded generalized $V$-module $W$, we give $W$ a structure of
graded $A^{\infty}(V)$-module and we introduce an $A^{\infty}(V)$-bimodule
$A^{\infty}(W)$ and an $A^{N}(V)$-bimodule $A^{N}(W)$. We prove that the space
of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ for
lower-bounded generalized $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$ is
isomorphic to the space
$\hom_{A^{\infty}(V)}(A^{\infty}(W_{1})\otimes_{A^{\infty}(V)}W_{2}, W_{3})$.
Assuming that $W_{2}$ and $W_{3}'$ are equivalent to certain universal
lower-bounded generalized $V$-modules generated by their $A^{N}(V)$-submodules
consisting of elements of levels less than or equal to $N\in \mathbb{N}$, we
also prove that the space of (logarithmic) intertwining operators of type
$\binom{W_{3}}{W_{1}W_{2}}$ is isomorphic to the space of
$\hom_{A^{N}(V)}(A^{N}(W_{1})\otimes_{A^{N}(V)}\Omega_{N}^{0}(W_{2}),
\Omega_{N}^{0}(W_{3}))$.
