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We show that any pointed, preordered module map$\mathfrak{BF}_{\mathrm{gr}}(E) \to \mathfrak{BF}_{\mathrm{gr}}(F)$ betweenBowen-Franks modules of finite graphs can be lifted to a unital, graded,diagonal preserving $\ast$-homomorphism $L_\ell(E) \to L_\ell(F)$ between thecorresponding Leavitt path algebras over any commutative unital ring withinvolution $\ell$. Specializing to the case when $\ell$ is a field, weestablish the fullness part of Hazrat's conjecture about the functor fromLeavitt path $\ell$-algebras of finite graphs to preordered modules with orderunit that maps $L_\ell(E)$ to its graded Grothendieck group. Our constructionof lifts is of combinatorial nature; we characterize the maps arising from thisconstruction as the scalar extensions along $\ell$ of unital, graded$\ast$-homomorphisms $L_{\mathbb Z}(E) \to L_{\mathbb Z}(F)$ that preserve asub-$\ast$-semiring introduced here.

Lifting morphisms between graded Grothendieck groups of Leavitt path algebras