I) First of all, one should never use the [Dirac bra-ket notation](http://en.wikipedia.org/wiki/Bra-ket_notation) (in its ultimative version where an operator acts to the right on kets and  to the left on bras) to consider the definition of [adjointness]((http://en.wikipedia.org/wiki/Adjoint_operator)), since the notation was designed to make the adjointness property look like a mathematical triviality, which it is not. See also [this](http://physics.stackexchange.com/q/43069/2451) Phys.SE post.

II) Let us next recall the definition of the adjoint of a linear operator. Let there be a [Hilbert space](http://en.wikipedia.org/wiki/Hilbert_space) $H$ over a [field](http://en.wikipedia.org/wiki/Field_%28mathematics%29) $\mathbb{F}$, which could either be $\mathbb{R}$ or $\mathbb{C}$. In the complex case we will use the standard physicist's convention that the inner product $\langle \cdot | \cdot \rangle$ is conjugated $\mathbb{C}$-linear in the first entry, and $\mathbb{C}$-linear in the second entry.

Recall [Riesz' representation theorem](http://en.wikipedia.org/wiki/Riesz_representation_theorem): For each continuous $\mathbb{F}$-linear functional $f: H \to \mathbb{F}$ there exists a unique vector $u\in H$ such that
$$\tag{1} f(\cdot)~=~\langle u | \cdot \rangle.$$

Let $A:H\to H$ be a continuous$^1$ $\mathbb{F}$-linear operator. Let $v\in H$ be a vector. Consider the continuous $\mathbb{F}$-linear functional $$\tag{2} f(\cdot)~=~\langle v | A(\cdot) \rangle.$$ 
The value $A^{\dagger}v\in H$ of the adjoint operator $A^{\dagger}$ at the vector $v\in H$ $A$ is by definition the unique vector $u\in H$, guaranteed by Riesz' representation theorem, such that 
$$\tag{3} f(\cdot)~=~\langle u | \cdot \rangle.$$

In other words, 
$$\tag{4} \langle A^{\dagger}v | w \rangle~=~\langle u | w \rangle~=~f(w)=\langle v | Aw \rangle. $$

It is straightforward to check that the adjoint operator $A^{\dagger}:H\to H$ becomes an $\mathbb{F}$-linear operator as well. 

III) Now let us return to OP's question(v1) and consider the definition of the adjoint of an [antilinear](http://en.wikipedia.org/wiki/Antilinear_map) operator. The definition will rely on the complex version of Riesz' representation theorem. Let $H$ be a complex Hilbert space, and let  $A:H\to H$ an antilinear continuous operator. In this case, the above equations (2) and (4) should be replaced with

$$\tag{2'} f(\cdot)~=~\overline{\langle v | A(\cdot) \rangle},$$

and

$$\tag{4'} \langle A^{\dagger}v | w \rangle~=~\langle u | w \rangle~=~f(w)=\overline{\langle v | Aw \rangle}, $$

respectively. Note that $f$ is a $\mathbb{C}$-linear functional.

It is straightforward to check that the adjoint operator $A^{\dagger}:H\to H$ becomes an antilinear operator as well.  


  
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$^{1}$We will ignore subtleties with [unbounded operators](http://en.wikipedia.org/wiki/Unbounded_operator), domains, [selfadjoint extensions](http://en.wikipedia.org/wiki/Extensions_of_symmetric_operators), etc., in this answer.