TITLE: Comparing $\mathcal C$ and $\mathcal C^{\mathcal C}$ (where $\mathcal C$ is a category)
QUESTION [5 upvotes]: This is a followup to this question. (Matt Feller also mentioned this followup in a comment to the question linked to above.)
For any category $\mathcal C$ write $\mathcal C^{\mathcal C}$ for the category of endofunctors of $\mathcal C$, and write $[\operatorname{Ob}({\mathcal C})]$ for the collection of isomorphism classes of objects of $\mathcal C$. (Note that $[\operatorname{Ob}({\mathcal C})]$ is not necessarily a set.)
Let $\mathcal C$ be a category which is not equivalent to a category having exactly one object and one morphism.
Are the following statements necessarily true?
(1) There is no surjection $[\operatorname{Ob}(\mathcal C)]\to[\operatorname{Ob}(\mathcal C^{\mathcal C})]$.
(2) There is no injection $[\operatorname{Ob}(\mathcal C^{\mathcal C})]\to[\operatorname{Ob}(\mathcal C)]$.
A positive answer to at least one of the above questions would also answer the question linked to above. A negative answer to at least one of the above questions would also answer this older question. [Emil Jeřábek noticed this mistake.]
(We assume that we are working in ZFC.)

REPLY [5 votes]: If $\mathcal C$ is the category attached to the ordered set $(\mathbb R,\le)$, then $[\operatorname{Ob}(\mathcal C)]$ coincides with the set $\mathbb R$ and $[\operatorname{Ob}(\mathcal C^{\mathcal C})]$ coincides with the set of weakly increasing functions from $\mathbb R$ to $\mathbb R$. But it is well known that these two sets are equipotent: see for instance this answer of Brian M. Scott or this answer of mechanodroid.