I asked a closely related question on Mathematics Stackexchange but got no answer.

Let $\mathbf1$ be a category with exactly one object and one morphism, and, for any category $\mathcal C$, write $\mathcal C^{\mathcal C}$ for its category of endofunctors.

I previously asked if the equivalence $\mathcal C^{\mathcal C}\sim\mathcal C$ implies that $\mathcal C\sim\mathbf1$, 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 $\mathcal C^{\mathcal C}\sim\mathcal C\implies\mathcal C\sim\mathbf1$ 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 that $X$ is a singleton, and similarly for a surjective morphism $X\to\text{End}(X)$.

This suggests the following questions:

Let $\mathcal C$ be a category.

Question 1 Does the existence of a fully faithful functor $\mathcal C^{\mathcal C}\to\mathcal C$ imply $\mathcal C\simeq\mathbf1$?

Question 2 Does the existence of an essentially surjective functor $\mathcal C\to\mathcal C^{\mathcal C}$ imply $\mathcal C\simeq\mathbf1$?

Of course a positive answer to any of these questions would also solve the previously asked question.

[Edit: The comments refer to a previous version of the question.]