Is the category of class of all sets a concrete category? Since the class of all sets is not a set but is  a class, is the category of  class of all sets a concrete category?
(the only object is the class of all sets)

I'm assuming the situation is as follows: We have a category $\mathcal{C}$ with one object $*$ and for morphisms we take the collection of functors $Set \to Set$. For this to be a concrete category we would need a functor $\mathcal{C} \to Set$ such that the corresponding map on homs is injective. But the collection of morphisms $* \to *$ is not a set! To see this, consider the functor $Set \to Set$ that sends every object to a given set $S$, and morphisms to the identity $S \to S$. This gives us an injection $V \to \operatorname{Hom}(*,*)$, where $V$ is the universe of all sets. Since $V$ is not a set, we're done. 
A similar argument works if we take $\operatorname{Hom}(*,*)$ to be the collection of functors $Ord \to Ord$, from the category of ordinals to itself (I mention this since it was asked in the comments).
