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 $\mathbf C$.

Let $\map F f : \map F x \to \map F y$ be 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 $f \circ g = f \circ h$.

\(\ds f \circ g\)

\(=\)

\(\ds f \circ h\)

\(\ds \leadsto \ \ \)

\(\ds \map F f \circ \map F g\)

\(=\)

\(\ds \map F f \circ \map F h\)

Definition of Functor

\(\ds \leadsto \ \ \)

\(\ds \map F g\)

\(=\)

\(\ds \map F h\)

$\map F f$ is a monomorphism in $\mathbf D$

\(\ds \leadsto \ \ \)

\(\ds g\)

\(=\)

\(\ds h\)

Definition of Faithful Functor

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

$\blacksquare$