Pullback diagrams, syzygy finite classes and Igusa-Todorov algebras

For an abelian category $\mathcal{A}$, we define the category PEx($\mathcal{A}$) of pullback diagrams of short exact sequences in $\mathcal{A}$, as a subcategory of the functor category Fun($\Delta, \mathcal{A}$) for a fixed diagram category $\Delta$. For any object $M$ in ${\rm PEx}(\mathcal{A}),$ we prove the existence of a short exact sequence $0 {\to} K {\to} P {\to} M {\to} 0$ of functors, where the objects are in PEx($\mathcal{A}$) and $P(i) \in {\rm Proj(\mathcal{A})}$ for any $i \in \Delta$. As an application, we prove that if $(\mathcal{C}, \mathcal{D}, \mathcal{E})$ is a triple of syzygy finite classes of objects in $\mathrm{mod}\,\Lambda$ satisfying some special conditions, then $\Lambda$ is an Igusa-Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa-Todorov.