Gorenstein projective objects in comma categories

Let $$\mathcal {A}$$ and $$\mathcal {B}$$ be abelian categories and $${\mathbf {F}} :\mathcal {A}\rightarrow \mathcal {B}$$ an additive and right exact functor which is perfect, and let $$({\mathbf {F}},\mathcal {B})$$ be the left comma category. We give an equivalent characterization of Gorenstein projective objects in $$({\mathbf {F}},\mathcal {B})$$ in terms of Gorenstein projective objects in $$\mathcal {B}$$ and $$\mathcal {A}$$ . We prove that there exists a left recollement of the stable category of the subcategory of $$({\mathbf {F}},\mathcal {B})$$ consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in $$\mathcal {B}$$ and $$\mathcal {A}$$ . Moreover, this left recollement can be filled into a recollement when $$\mathcal {B}$$ is Gorenstein and $${\mathbf {F}}$$ preserves projectives.