Cotorsion pairs in comma categories

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