I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the the Cantor-Schroder-Bernstein property? That is, does for any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exists embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functor) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help!

[This post][1] is of relevance.

Following @Joel David Hamkins’ answer, assuming his counterexample can be extended to the weaker statement, it remains to prove that there is no biomorphic functor $\mathcal{F}:(\mathbb{Q},≥)→(\mathbb{Q}^{\geq 0},≥)$ or, on the other side, that there is no bimorphic functor $\mathcal{F}:(\mathbb{Q}^{\geq 0},≥)→(\mathbb{Q},≥)$. [1]: When does Cantor-Bernstein hold?.