What is the Stinespring representation of the adjoint of a channel?

For any completely positive trace nonincreasing map $N_{Arightarrow B}$, the adjoint map is the unique completely positive linear map $N^dagger_{Brightarrow A}$ that satisfies

$$langle N^dagger(sigma), rhorangle = langle sigma, N(rho)rangle$$

for all linear operators $sigma in mathcal{L}(mathcal{H}_B)$ and $rho in mathcal{L}(mathcal{H}_A)$.

Let $V_{Arightarrow BE}$ be any isometry such that $text{Tr}_E(Vrho V^dagger) = N(rho)$. This is the Stinespring representation of any completely positive map. Since $N^dagger$ is also a completely positive map, it also has a Stinespring representation.

Question: Given $V$, can one write down the Stinespring representation of $N^dagger$? Naively taking the transpose conjugate of $V$ to write down something like

$$text{Tr}_E(V^daggersigma V) = N^dagger(sigma)$$

doesn’t even make sense since $V^dagger_{BErightarrow A}$ whereas the isometry we are after should go from $B$ to $AE’$.