about the definition of image and coimage in Milne's class field theory I begin to study the cohomology. I read Milne's course note Class Field Theory Chapter 2 and study the definition of abelian category. I can't understand the definition of image and coimage(the sentence in the picture).Milne's book definition of image and coimage.
What is a  kernel of a cokernel of $\alpha$?
I think cokernel is defined as an object in this page. What is a kernel of an object? 
 A: This is a somewhat standard abuse of language. If one is being precise, a kernel is category-theoretically not an object, but a morphism (and one sometimes calls the domain of this morphism the kernel). At the end of the first paragraph, what should really be said is that $\beta : A\to B$ is the kernel of $\alpha$, and that $\alpha : B\to C$ is the cokernel of $\beta$. Precisely:
Definition: Let $\mathsf{C}$ be a category with a zero object (and hence a zero morphism between any two objects), let $B$ and $C$ be objects of $\mathsf{C}$, and let $\alpha : B\to C$ be a morphism in $\mathsf{C}$. A kernel of $f$ is a morphism $\beta : A\to B$ such that $\alpha\circ \beta = 0$, and given any morphism $\beta' : A'\to B$ such that $\alpha\circ \beta' = 0$, there exists a unique morphism $u : A'\to A$ such that $\beta\circ u = \beta'$. (There's a nice diagram of this property on the wikipedia page.)
A cokernel can be defined by reversing all the arrows in the definition. As I mentioned above, a standard abuse of language is to call the object $A$ the kernel of $\alpha$ instead of the morphism $\beta : A\to B$. This is typically done when the morphism $\beta : A\to B$ is not ambiguous.
In Milne's notes, he is really saying that the image of $\alpha : A\to B$ is the kernel of the cokernel of $\alpha$, all considered as morphisms. He denotes the cokernel of $\alpha$ by the morphism $B\to C$, and you must take the kernel of this morphism to define the image of $\alpha$ (note that the image of $\alpha$ is again a morphism, although often one will think of it as an object $I$ that comes with a distinguished morphism $\iota : I\to B$).
Until you get comfortable with this change of perspectives (from morphisms to objects back to morphisms), I suggest translating everything into morphism language to be careful and precise.
A: Say you have $f : A \to B$. The kernel of $f$ is $\iota : \ker(f) \rightarrowtail A$. That is, it's a subobject which means it consists of not only the object $\ker(f)$ but the inclusion of that object back into $A$, which I've called $\iota$ in this case.  $\iota$ is a perfectly good morphism itself, so you can ask for its cokernel, $e : \ker(f) \twoheadrightarrow \text{coker}(\iota)$. Often we'll write something slightly ambiguous like $\text{coker}(\ker(f))$.  $\text{coker}(\iota)$ is now $\text{coim}(f)$.  The image case is completely dual.  
It is a common abuse of notation to "neglect" the relevant arrows that are part of the definition of a subobject or quotient or various other objects defined by universal properties, and to just talk about the object part with the arrows being "understood".  For example, we say $A\times B$ is the product of $A$ and $B$, but it's actually the triple $(A\times B,\pi_1, \pi_2)$ which is the product.  The same object with different associated arrows leads to a different product, e.g. $(A\times B, \pi_2,\pi_1)$ behaves like $B\times A$.
