$\tau$-cluster morphism categories of factor algebras

We take a novel lattice-theoretic approach to the $\tau$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I:\mathfrak{T}(A)\to \mathfrak{T}(A\mathrm{/}I)$. If $\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $\tau$-cluster morphism category up to equivalence.