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.

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)?