# Possible similarities between a category and its category of endofunctors

The poset variant of this question is posted separately here.

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 (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)?

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.

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

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

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

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