From P309 of Rotman's Intro to Homological Algebra
Thm 5.91. If A is an abelian category, then Sh(X, A) is an abelian category.
The proof boils down to showing that monomorphisms $\varphi$ are kernels and epimorphisms $\psi$ are cokernels.
Rotman states that there is an exact sequence of sheaves
0 $\to$ K $\xrightarrow{\iota}$ F $\xrightarrow{\varphi}$ G $\xrightarrow{\nu}$ K' $\to$ 0.
If $\varphi$ is monic, then ker $\varphi$ = 0. And so 0 $\to$ F $\xrightarrow{\varphi}$ G $\xrightarrow{\nu}$ K' is an exact sequence in Sh(X, A). Hence there is an exact sequence 0 $\to$ F$_x$ $\xrightarrow{\varphi_x}$ G$_x$ $\xrightarrow{\nu_x}$ K'$_x$ in A; that is im$\varphi$$_x$ = ker $\nu_x$ for all x. Therefore, $\varphi$ = ker $\nu$.
I'm not sure how the last "therefore" conclusion was reached. is it because $\varphi$ can be factorized as a composition of im$\varphi$ and cokernel(kernel$\varphi$)? I'm new to this, any help would be appreciated!