Faithful Functor Reflects Monomorphisms

Theorem
Let $\mathbf C$ and $\mathbf D$ be categories.

Let $F: \mathbf C \to \mathbf D$ be a faithful functor.

Let $x$ and $y$ be objects in $\mathbf C$.

Let $f: x \to y$ be a morphism in $\mathbf C$.

Assume that $\map F f : \map F x \to \map F y$ is a monomorphism in $\mathbf D$.

Then $f$ is a monomorphism in $\mathbf C$.

Proof
Let $z$ be an object in $\mathbf C$.

Let $g: z \to x$ and $h: z \to x$ be morphisms in $\mathbf C$ such that $f \circ g = f \circ h$.

Hence $f$ is a monomorphism in $\mathbf C$.