Triangular matrix categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $\mathbf {{\varLambda }}=\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M {U} \end {smallmatrix}\right ]$ initiated in León-Galeana et al. (2022). First, given a additive category $\mathcal {C}$ and an ideal {I}_{{\mathscr{B}}}$ , prove well known result that there is canonical recollement We show between functor can induce new categories, generalization by Chen Zheng (J. Algebra, 321 (9), 2474–2485 2009, [Theorem 4.4]). case dualizing K-varieties restrict obtained to finitely presented functors. Given variety describe maps $\text {mod}(\mathcal {C})$ as modules over its Auslander-Reiten sequences contravariantly finite subcategories, particular generalize several results from Martínez-Villa Ortíz-Morales (Inter. J 5 (11), 529–561 2011). Finally, due Smalø (2011, 2.1]), which give us way construct functorially subcategories {Mod}\Big (\left ]\Big )$ those {Mod}(\mathcal {T})$ {U})$ .