  Let $n$ be a positive integer. We show that an $n$-extension closed
subcategory of an $n$-exangulated category naturally inherits an
$n$-exangulated structure through restriction of the ambient $n$-exangulated
structure. Furthermore, we show that a strong version of the Obscure Axiom
holds for $n$-exangulated categories, where $n \geq 2$. This allows us to
characterize $n$-exact categories as $n$-exangulated categories with monic
inflations and epic deflations. We also show that for an extriangulated
category condition (WIC), which was introduced by Nakaoka and Palu, is
equivalent to the underlying additive category being weakly idempotent
complete. We then apply our results to show that $n$-extension closed
subcategories of an $n$-exact category are again $n$-exact. Furthermore, we
recover and improve results of Klapproth and Zhou.
