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.)