$tau$-perpendicular wide subcategories

Let $Lambda$ be a finite-dimensional algebra. A wide subcategory of $mathsf{mod}Lambda$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of $mathsf{mod}Lambda$ 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 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.