Consider the category of smooth manifolds $\mathcal{M}$ with the morphisms being the smooth maps. Consider the category of manifolds with a distinguished point $\mathcal{M}_o.$ The objects of $\mathcal{M}_o$ are ordered pairs $(M,x)$ such that $ x \in M$ and morphisms between $(M_1,x)$ and $(M_2,y)$ are smooth maps $f:M_1 \to M_2$ such that $ f(x)=y.$
Is it true that there are no functors $F$ from $\mathcal{M}$ to $\mathcal{M}_o$ such that $$ F(M)=(M,x)$$ for some $x \in M?$