  In this paper, given a module $W$ for a vertex operator algebra $V$ and a
nonzero complex number $z$ we construct a canonical (weak) $V\otimes V$-module
${\cal{D}}_{P(z)}(W)$ (a subspace of $W^{*}$ depending on $z$). We prove that
for $V$-modules $W, W_{1}$ and $W_{2}$, a $P(z)$-intertwining map of type
${W'\choose W_{1}W_{2}}$ ([H3], [HL0-3]) exactly amounts to a $V\otimes
V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. Using
Huang and Lepowsky's one-to-one linear correspondence between the space of
intertwining operators and the space of $P(z)$-intertwining maps of the same
type we obtain a canonical linear isomorphism from the space
${\cal{V}}^{W'}_{W_{1}W_{2}}$ of intertwining operators of the indicated type
to $\Hom_{V\otimes V}(W_{1}\otimes W_{2},{\cal{D}}_{P(z)}(W))$. In the case
that $W=V$, we obtain a decomposition of Peter-Weyl type for
${\cal{D}}_{P(z)}(V)$, which are what we call the regular representations of
$V$.
