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M. Herschend, Y. Liu, H. Nakaoka introduced $n$-exangulated categories, which are a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. This paper consists of two results on $n$-exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$-exangulated category.

Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T:𝒜\to ℬ$ a left exact additive functor. Then one has a comma category $\left(ℬ\downarrow T\right)$. It is shown that if $T:𝒜\to ℬ$ is $𝒳$-exact, then ${(}^{\perp }𝒳,𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and ${(}^{\perp }𝒴,𝒴)$) is a (hereditary) cotorsion pair in $ℬ$ if and only if $\left(\left(\genfrac{}{}{0pt}{}{{}^{\perp }𝒳}{{}^{\perp }𝒴}\right)\right),\langle 𝐡(𝒳,𝒴)\rangle )$ is a (hereditary) cotorsion pair in $\left(ℬ\downarrow T\right)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping classes...