Let A and B are two objects in a category $\mathcal{C}$ and $u \in Hom(A,B)$. We say $u $ is a monomorphism if and only if for any $X \in Obj(\mathcal{C})$ and $v \in Hom(X,A)$, the map $f:Hom(X,A)\rightarrow Hom(X,B)$ given by $f(v)=uv $ is a monomorphism.
My question:
Is there a same kind of if and only if statement for category epimorphism ?
Yes (though every definition can be viewed as an "if and only if statement"). The definition of an epimorphism in a category $\mathcal{C}$ is an arrow $f:X\to Y$ such that whenever $Z$ is an object and $u,v:Y\to Z$ are arrows, $u\circ f=v\circ f$ implies $u=v$. In other words, $f:X\to Y$ is an epimorphism if and only if for all objects $Z$ the map $f^*:\mathrm{Hom}_{\mathcal{C}}(Y,Z)\to\mathrm{Hom}_{\mathcal{C}}(X,Z)$ given by $f^*(u)=u\circ f$, is injective.