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Following Mitchell’s philosophy, in this paper we define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two preadditive categories and and we construct the triangular matrix category \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf {{\varLambda }}:=\left [\begin {array}{ll} \mathcal {T} & 0 \\ M & \mathcal {U} \end {array}\right ]$\end{document}. First, we prove that Mod(Λ) is equivalent to a comma category which is induced by a functor . One of our main results is that if and are dualizing K-varieties and satisfies certain conditions then \documentclass[12pt …