I want to prove that in the category of R-Modules, all monomorphisms are equalizers. We start by assuming that if $f : A \to B$ is mono and if $f$ equalizes $g : B \to C$ and $h : B \to C$ (i.e. $ g \circ f = h \circ f$) then for every $f' : A' \to B $ that equalizes $g,h$ there is a unique $f : A' \to A$ such that $ f \circ u = f'$.
It is clear to me that if we can establish a module morphism $u$ from $A'$ to $A$ such that $ f \circ u = f'$ it follows from the fact that $f$ is monic that it will be unique. It is not clear to me how to define that morphism. Any help would be appreciated.