$\tau$-perpendicular wide subcategories

Let $\Lambda$ be a finite-dimensional algebra. A wide subcategory of mod$\Lambda$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of mod$\Lambda$ arising from $\tau$-tilting reduction are precisely the Serre subcategories of left finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under intersections and under further $\tau$-tilting reduction. This leads to a natural way to extend the definition of the "$\tau$-cluster morphism category" of $\Lambda$ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan--Marsh in the $\tau$-tilting finite case and by Igusa--Todorov in the hereditary case.