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

Well, basically on the right direction, but not exactly the right track. You should start it over and simplify.
You miss the observation that the domain of the kernel is going to be the zero object itself, and not $A$.
Claim: The kernel of a monic $f:A\to B$ will be the unique arrow $Z\to A$. 
Proof: (your turn)
