This question was already asked on MSE.
In CAT the Yoneda embedding $C^{\text{op}} \stackrel{y_C}{\to} \text{hom}(C, \text{Set})$, induces a map $$\text{hom}(\text{hom}(C, \text{Set}), \text{Set}) \stackrel{y^*_C}{\to} \text{hom}(C^{\text{op}}, \text{Set})$$
Thus, if we call $T = \text{hom}(\_, \text{Set})$. We have a map (which is not even dinatural): $$\_^{\text{op}} \stackrel{\iota}{\to} T$$ and a kind of multiplication,
$$T^2 \stackrel{\mu}{\to} T \circ \_^{\text{op}}. $$
Is there a way to say that $T$ is contravariant monad on CAT?! I looked online for such a notion and I did not find anything.
Note, even if $\iota$ is not dinatural, i.e. $f^*y_Hf \neq y_G$, for a functor $G \stackrel{f}{\to} H$, looking at CAT as a 2-category, there is a $2$-cell $$y_G \Rightarrow f^*y_Hf.$$
What does it mean? If we look at the presheaf construction as a kind of power set the map $y_G \Rightarrow f^*y_Hf$ is witnessing that $\{a\} \subset f^{-1}\{f(a)\}$ for a function $f: A \to B$ between sets and an element $a \in A$.
Another way to say it is that the $2$-cell is $y_G \Rightarrow f^*y_Hf$ is individuated by the universal property of $f^* \cong \text{Lan}_{y_Hf}(y_G)$.