I asked this question on Mathematics Stackexchange but got no answer.

For any category $\mathcal C$ write $\mathcal C^{\mathcal C}$ for its category of endofunctors.

EDIT In view of the downvotes and the comments I'm adding this edit to try to clarify the motivation.

I previously asked if the equivalence $\mathcal C^{\mathcal C}\sim\mathcal C$ implies that $\mathcal C$ is equivalent to a category with exactly one object and one morphism, but got no complete answer.

In the meantime I realized that Theorem 3 in Complete lattices and the generalized Cantor theorem by Roy O. Davies, Allan Hayes and George Rousseau, published in Proc. Amer. Math. Soc. 27 (1971), 253–258, link, shows that the implication $\mathcal C^{\mathcal C}\sim\mathcal C\implies\mathcal C\sim1$ does hold for posets.

But the theorem in question is much stronger! Indeed it shows that, for a poset $X$, the existence of an injective morphism $\text{End}(X)\to X$ implies $|X|=1$, and similarly for a surjective morphism $X\to\text{End}(X)$, so that we have an equivalence between four natural properties of a poset.

There is an obvious translation of these four properties for categories. I could have asked "are these four properties of a category equivalent?", but this would have been a partial duplicate of the previous question, and I thought it would be clearer to list precisely the implications I am not able to prove or disprove. The price to pay was a post with a lot of questions, and somewhat unappealing. END OF EDIT

Consider the following properties a category $\mathcal C$ may or may not have:

(P1) $\mathcal C$ is equivalent to a category with exactly one object and one morphism,

(P2) $\mathcal C^{\mathcal C}$ is equivalent to $\mathcal C$,

(P3) there is a fully faithful functor $\mathcal C^{\mathcal C}\to\mathcal C$,

(P4) there is an essentially surjective functor $\mathcal C\to\mathcal C^{\mathcal C}$.

Clearly (P1) implies (P2), and (P2) implies (P3) and (P4): $$ \begin{matrix} &&1\\ &&\downarrow\\ &&2\\ &\swarrow&&\searrow\\ 3&&&&4. \end{matrix} $$ Denote by (Qij) the question "Does (Pi) imply (Pj)?".

Question (Q21) was asked here. Let us ask now:

Question (Q31) Does (P3) imply (P1)?

Question (Q41) Does (P4) imply (P1)?

Question (Q32) Does (P3) imply (P2)?

Question (Q42) Does (P4) imply (P2)?

Question (Q34) Does (P3) imply (P4)?

Question (Q43) Does (P4) imply (P3)?

For completeness sake let us ask also

Question (Q34,1) Do (P3) and (P4) imply (P1)?

Question (Q34,2) Do (P3) and (P4) imply (P2)?

By Theorem 3 in Complete lattices and the generalized Cantor theorem by Roy O. Davies, Allan Hayes and George Rousseau, published in Proc. Amer. Math. Soc. 27 (1971), 253–258, link, (P1), (P2), (P3) and (P4) are equivalent if $\mathcal C$ is a poset.

There are very natural categories for which I don't know which of the properties (P2), (P3) or (P4) hold. Let (Qi,$\mathcal C$) be the question "Does the category $\mathcal C$ have Property (Pi)?". In this post, Question (Q2,$\mathsf{Set}^{\mathsf{Set}}$) was asked and Question (Q2,$\mathsf{Set}$) was answered negatively.

Question (Q3,$\mathsf{Set}$) Does $\mathsf{Set}$ have Property (P3)?

Question (Q4,$\mathsf{Set}$) Does $\mathsf{Set}$ have Property (P4)?

Question (Q3,$\mathsf{Set}^{\mathsf{Set}}$) Does $\mathsf{Set}^{\mathsf{Set}}$ have Property (P3)?

Question (Q4,$\mathsf{Set}^{\mathsf{Set}}$) Does $\mathsf{Set}^{\mathsf{Set}}$ have Property (P4)?