Given a category $\mathcal C$ together with a monad $T$ on $\mathcal C$, we get an adjunction $$\mathcal C^T(T,)\cong \mathcal C(,\mathrm{For}).$$
Is the same true for 2monads on a 2category?
Given a category $\mathcal C$ together with a monad $T$ on $\mathcal C$, we get an adjunction $$\mathcal C^T(T,)\cong \mathcal C(,\mathrm{For}).$$ Is the same true for 2monads on a 2category? 


As Todd says, there are several flavours of 2monad. If you are interested in strict 2monads, strict algebras for these, and strict morphisms, then yes you have an adjunction (even an enriched adjunction) as usual. If you mean something weaker, then you will have something weaker than an adjunction. In particular, if you are considering pseudomorphisms of algebras  those which preserve the structure only up to (suitably coherent) isomorphism  then you'll have an equivalence between the category of algebra morphisms from a free algebra $TX$ to an algebra $B$ and the category of morphisms in the base 2category from $X$ to (the underlying object of) $B$. So rather than an adjunction you'll get some sort of biadjunction. See Corollary 5.6 of the BlackwellKellyPower paper for the case of strict monads, strict algebras, and pseudomorphisms, which is in fact the most important case for many purposes. An important aspect of this is that free algebras $TX$ are flexible, which means among other things that any pseudomorphism from $TX$ to $B$ is isomorphic to a strict one. This is false for a general algebra $A$ in place of $TX$. 


Well, yes. There are several flavors of 2monad, as explained in this nLab article, but I'm guessing you mean something like a pseudomonad $T$, and the 2category $C^T$ of pseudoalgebras over $T$ (with the natural isomorphism replaced by a pseudonatural equivalence). The references in the nLab article may have what you need; I'd look specifically at the article by Blackwell, Kelly, and Power (I don't have the article at hand to confirm this). 

