kernel of a monic morphism Problem
Suppose $\mathscr{C}$ is an arbitrary category with zero object. $A$ and $B$ are two objects of $\mathscr{C}$. Let $f\in Mor_\mathscr{C}(A,B)$. It's given that $f$ is monic.
I need to show that $f$ has a kernel. 
My claim Object $A$ with morhism $\iota$(will be def'd below) is the kernel of $f$. I did the followings: 
1- Suppose $Z$ is zero object. I defined the morhisms: 
$ \phi_{a} : Z \rightarrow  A $ ,
$ \phi_{b} : Z \rightarrow  B $ 
$ \psi_{a} : A \rightarrow  Z $ , 
$ \psi_{a} : B \rightarrow  Z $ 
Since Z is both initial and terminal these morphisms are in some sense "unique". 
2- I defined $\iota=\phi_{a} \circ \psi_{a}$. I showed that $f \circ \iota = 0_{AB}$ ie. the zero morhism from $A$ to $B$. 
3- Suppose there is an object $D$ with morphism $g: D \rightarrow A$ such that $f \circ g = 0_{DB}$. I showed that for every morphism $h: D \rightarrow A$ we have $g = \iota \circ h$.
4- I failed to prove uniqueness of h.

My question Could you please tell me that if i am in the right path or not? If i am, how can i fix the 4th step? Thanks

Having a zero object in your category gives you a (unique) zero morphism $A\to B$ obtained as composition $A\to 0\to B$ for any two objects $A,B\in\cal C$.
This gives on its own right a practical definition for $\ker (f)$: it is the equalizer of the arrow $f$ and the zero map; you can have fun in finding that this is precisely the right definition for "what is sent to $0$ by $f$" (for example, what if $\cal C$ is the category of modules over a ring, or more simply vector spaces, or less trivially pointed sets?
Once you know this, and the fact that $\ker(f)$, whenever it exists, is unique up to a unique iso, your problem boils down to showing that the unique arrow $0\to A$ from zero object to the domain of $f$ serves as the kernel of any monic arrow $f\colon A\to B$.
