  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 $\mathrm{tors } A$. Using the
lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I:
\mathfrak{T}(A) \to \mathfrak{T}(A/I)$ and if $\mathrm{tors } A$ is finite an
inclusion $\mathcal{I}: \mathfrak{T}(A/I) \to \mathfrak{T}(A)$. We characterise
when these functors are full, faithful and adjoint. As a consequence we find a
new family of algebras for which $\mathfrak{T}(A)$ admits a faithful group
functor.
