I asked [this question](https://math.stackexchange.com/q/3302519/660) 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](https://mathoverflow.net/q/334266/461) 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](http://www.ams.org/journals/proc/1971-027-02/S0002-9939-1971-0268091-0/S0002-9939-1971-0268091-0.pdf), 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](https://mathoverflow.net/q/334266/461). 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](http://www.ams.org/journals/proc/1971-027-02/S0002-9939-1971-0268091-0/S0002-9939-1971-0268091-0.pdf), (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](https://math.stackexchange.com/q/3295737/660), 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)?