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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 additive categories and and we construct the triangular matrix category $\mathbf{\Lambda}:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$. First, we prove that there is an equivalence . One of our main results is that if and are dualizing -varieties and satisfies certain conditions then $\mathbf{\Lambda}:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ is a dualizing variety (see theorem 6.10). In particular, has Auslander-Reiten sequences. Finally, we apply the theory developed in this paper to quivers and give a generalization of the so called one-point extension algebra.