Let $mathscr{F}$ be an $(n+2)$-angulated Krull-Schmidt category and $mathscr{A} subset mathscr{F}$ an $n$-extension closed, additive and full subcategory with $operatorname{Hom}_{mathscr{F}}(Sigma_n mathscr{A}, mathscr{A}) = 0$. Then $mathscr{A}$ naturally carries the structure of an $n$-exact category in the sense of Jasso, arising from short $(n+2)$-angles in $mathscr{F}$ with objects in $mathscr{A}$ and there is a binatural and bilinear isomorphism $operatorname{YExt}^{n}_{(mathscr{A},mathscr{E}_{mathscr{A}})}(A_{n+1},A_0) cong operatorname{Hom}_{mathscr{F}}(A_{n+1}, Sigma_n A_{0})$ for $A_0, A_{n+1} in mathscr{A}$. For $n = 1$ this has been shown by Dyer and we generalize this result to the case $n > 1$. On the journey to this result, we also develop a technique for harvesting information from the higher octahedral axiom (N4*) as defined by Bergh and Thaule. Additionally, we show that the axiom (F3) for pre-$(n+2)$-angulated categories, introduced by Geiss, Keller and Oppermann and stating that a commutative square can be extended to a morphism of $(n+2)$-angles, implies a stronger version of itself.
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