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We prove that a monomorphic functor $F:\mathbf{Comp}\to\mathbf{Comp}$ with finite supports isepimorphic, continuous, and its maximal $\emptyset$-modification $F^\circ$ preserves intersections. This implies that a monomorphic functor $F:\mathbf{Comp}\to\mathbf{Comp}$ of finite degree $\deg F\leq n$ preserves (finite-dimensional) compact ANRs if the spaces $F\emptyset$, $F^\circ\emptyset$ and $Fn$ are finite-dimensional ANRs. This improves a knownresult of Basmanov.