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We prove the equality $\cat(\phi)=\cd(\phi)$ for epimorphisms $\phi:\Gamma\to\Lambda$ between torsion-free, finitely generated almost nilpotent groups$\Gamma$ and $\Lambda$.  In addition, we prove the equality $\cat(\phi)=\cd(\phi)$ for homomorphisms$\phi:\Gamma\to \Lambda$ between torsion-free, finitely generated virtuallynilpotent groups.

On the LS-category of homomorphism of almost nilpotent groups