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 this 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 $$y_G \Rightarrow f^*y_Hf.$$
I am receiving some interesting comments on Overflow.