Is the unique subobject also mono?

Question

Suppose $u : S → A$ and $v : T → A$ with codomain $A$ are monomorphisms, we write $u ≤ v$ if $u$ factors through $v$—that is, if there exists $φ : S → T$ such that $u = v ∘ φ$. I understand that such a $\varphi$ is unique. But is it also a monomorphism? Why?

Answer 1

You need to know if $\varphi\circ x = \varphi\circ y \implies x=y$. Post-compose both sides of the antecedent with $v$.