Let \(G\) be a finite group. In the paper under review, the author constructs a functor \(\mathcal F\) from the category \(\mathbb{C} G\)-mod of finitely generated \(\mathbb{C} G\)-modules to the homotopy category \({\mathcal K}^b(\mathbb{C} G\text{-mon})\) of bounded chain complexes in the category \(\mathbb{C} G\)-mon. The objects of \(\mathbb{C} G\)-mon are finitely generated monomial \(\mathbb{C} G\)-modules, together with a fixed \(G\)-equivariant decomposition into 1-dimensional subspaces. For \(V\in\mathbb{C} G\)-mod, the chain complex \({\mathcal F}(V)\) can be viewed as a projective resolution of \(V\) in a suitable functor category. It has homology concentrated in degree 0 and isomorphic to \(V\). Moreover, the Lefschetz invariant of \({\mathcal F}(V)\), considered as an element in the Grothendieck ring of \(\mathbb{C} G\)-mon, coincides with the Canonical Brauer Induction Formula for \(V\), constructed by the author in an earlier paper [Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)]. In this way, the Canonical Brauer Induction Formula can be lifted to a functor \({\mathcal F}\colon\mathbb{C} G\text{-mod}\to{\mathcal K}^b(\mathbb{C} G\text{-mon})\). In the paper, the author also considers more general coefficient rings.