  We give a complete and self-contained exposition of the $J$-tame inflation
lemma: Given any tame almost complex structure $J$ on a symplectic $4$-manifold
$(M,\omega)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$,
it is always possible to construct a deformation of symplectic forms $\omega_t$
in classes $[\omega_t]=[\omega]+t\mathrm{PD}{Z}$, for $0\leq t$ less than an
upper bound $0<T$ that only depends on the self-intersection $Z\cdot Z$. The
original proofs of this fact make the unwarranted assumption that one can find
a family of normal planes along $Z$ that is both $J$ invariant and
$\omega$-orthogonal to $TZ$ -- which amounts, in effect, to assuming the
compatibility of $J$ and $\omega$ along $Z$. We explain how the original
constructions can be adapted to avoid this assumption when $Z$ has nonpositive
self-intersection, and we discuss the difficulties with this line of argument
in general to establish the full inflation when $Z$ has positive
self-intersection. We overcome this problem by proving a `preparation lemma',
which states that prior to inflation, one can isotop $\omega$ within its
cohomology class to a new form that still tames $J$ and which is compatible
with $J$ along the submanifold $Z$.
