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 $V$.
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