  Assume the existence of sufficent large cardinals. Let $M_{\mathrm{sw}n}$ be
the minimal iterable proper class $L[E]$ model satisfying "there are
$\delta_0<\kappa_0<\ldots<\delta_{n-1}<\kappa_{n-1}$ such that the $\delta_i$
are Woodin cardinals and the $\kappa_i$ are strong cardinals". Let
$M=M_{\mathrm{sw}2}$. We identify an inner model $\mathscr{V}_2^M$ of $M$,
which is a proper class model satisfying "there are 2 Woodin cardinals", and is
iterable both in $V$ and in $M$, and closed under its own iteration strategy.
The construction also yields significant information about the extent to which
$M$ knows its own iteration strategy. We characterize the universe of
$\mathscr{V}_2^M$ as the mantle and the least ground of $M$, and as
$\mathrm{HOD}^{M[G]}$ for $G\subseteq\mathrm{Coll}(\omega,\lambda)$ being
$M$-generic with $\lambda$ sufficiently large. These results correspond to
facts already known for $M_{\mathrm{sw}1}$, and the proofs are an elaboration
of those, but there are substantial new issues and new methods used to handle
them.
