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}$$)forafixeddiagramcategory$\Delta$\mathrm{.}Foranyobject$M$in${\rm PEx}($\mathcal{A}$),$weprovetheexistenceofashortexactsequence$0 {\to} K {\to} P {\to} M {\to} 0$ of functors, where the objects are in PEx($\mathcal{A}$) and $P(i)\in \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(\mathcal{A})$ for any $i \in \Delta$\mathrm{.}Asanapplication,weprovethatif$($\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$)$isatripleofsyzygyfiniteclassesofobjectsin$$\mathrm{m}\mathrm{o}\mathrm{d}$,\Lambda$satisfyingsomespecialconditions,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.