In this paper, we are going to look at the $c$-nilpotent multiplier of a group $G$, ${\\\\\\\\\\\\\\\\cal N}_cM(G)$, as a functor from the category of all groups, ${\\\\\\\\\\\\\\\\cal G}roup$, to the category of all abelian groups, ${\\\\\\\\\\\\\\\\cal A}b$, and focusing on some functional properties of it. In fact, by using some results of the first author and others and finding an explicit formula for the $c$-nilpotent multiplier of a finitely generated abelian group, we try to concentrate on the commutativity of the above functor with the two famous functors Ext and Tor.

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