This is a followup to [this question](https://mathoverflow.net/q/334266/461). (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](https://mathoverflow.net/q/211489/461)~~. [Emil Jeřábek noticed this mistake.]


(We assume that we are working in ZFC.)



Comparing $\mathcal C$ and $\mathcal C^{\mathcal C}$ (where $\mathcal C$ is a category)
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](https://math.stackexchange.com/a/278718/660) or [this answer of mechanodroid](https://math.stackexchange.com/a/2850143/660).


