The optimal way to concretize the opposite category of sets

Question

Let $\mathrm{Set}^{\mathrm{op}}$ be the category opposite to the category of sets. Does there exist a faithful functor $F:\mathrm{Set}^{\mathrm{op}}\rightarrow \mathrm{Set}$ such that $|F(X)|\leq |X|$ for all $X\in Obj(\mathrm{Set}^{\mathrm{op}})$ of infinite cardinality? I think the power set functor (with arrows going in the appropriate direction) gives an example of a faithful functor with $|F(X)|=2^{|X|}$. If it matters, assume the axiom of choice is on.

Answer 1

No. This would imply $$\kappa^\lambda\leq\lambda^\kappa$$ for any infinite cardinals $\kappa$ and $\lambda$, since $F$ would need to map all the functions $\lambda\to\kappa$ to distinct functions $F(\kappa)\to F(\lambda)$ and there are only $|F(\lambda)|^{|F(\kappa)|}\leq \lambda^\kappa$ such functions. But this is is false: for instance, if $\kappa$ is any infinite cardinal and $\lambda=2^\kappa$, then $$\kappa^\lambda=2^{2^\kappa}>2^\kappa=\lambda^\kappa.$$