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 2-monads on a 2-category?



Reference request: 2-Monads and 2-Adjunctions
As Todd says, there are several flavours of 2-monad. 


If you are interested in strict 2-monads, 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 2-category 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 Blackwell-Kelly-Power 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 $TX$.


