Triangular Matrix Categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $$\mathbf{\Lambda }$=\left[ \begin{smallmatrix} $\mathcal{T}$ & 0 \ M & $\mathcal{U}$ \end{smallmatrix}\right]$initiatedin[21]\mathrm{.}First,givenanadditivecategory$$\mathcal{C}$$andanideal$$\mathcal{I}${$\mathcal{B}$}$in$$\mathcal{C}$$, we prove a well known result that there is a canonical recollement $\xymatrix{$\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{C}$/$\mathcal{I}${$\mathcal{B}$})\ar[r]{} & $\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{C}$)\ar[r]{}\ar@<-1ex>[l]{}\ar@<1ex>[l]{} & $\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{B}$)\ar@<-1ex>[l]{}\ar@<1ex>[l]{}}$. We show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in [11, theorem 4.4]. In the case of dualizing $K$-varieties we can restrict the recollement we obtained to the categories of finitely presented functors. Given a dualizing variety $\mathcal{C}$, we describe the maps category of $\mathrm{m}\mathrm{o}\mathrm{d}(\mathcal{C})$ as modules over a triangular matrix category and we study its Auslander-Reiten sequences and contravariantly finite subcategories, in particular we generalize several results from [24]. Finally, we prove a generalization of a result due to {Smal\o} ([35, Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category $$\mathrm{M}\mathrm{o}\mathrm{d}$\Big(\left[ \begin{smallmatrix} $\mathcal{T}$ & 0 \ M & $\mathcal{U}$ \end{smallmatrix}\right]\Big)$fromthoseof$$\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{T}$)$and$$\mathrm{M}\mathrm{o}\mathrm{d}$($\mathcal{U}$)$.