A Monomorphism in a category is an epimorphism in opposite category.

Question

I am studying Emily Reihl's Category theory in context:

Let $f: x \rightarrow y$ be a monomorphism in a category. Then, for any parallel morphisms $h,k : w \rightrightarrows x, fh=fk \implies h=k $.

In the opposite category (where arrows gets reversed), we will have $ w \stackrel {h,k} \leftleftarrows x \xleftarrow{f} y$. How do we show that if $hf = kf$, then $h=k$ hold true?

Answer 1

For the sake of convenience, let's use the following notation in order to distinguish the objects and arrows in dual (also called oppsite) category. Objects in $C^{op}$ by $\overline{A}$ and morphisms by $\overline{f}$.

Then, the diagram looks like: $w \stackrel {h,k} \rightrightarrows x \xrightarrow{f} y$ in $C$ and, $\overline{w} \stackrel { \overline{h},\overline{k}} \leftleftarrows \overline{x} \xleftarrow{ \overline{f}} \overline{y}$ in $C^{op}$.

Let $\overline{h}\overline{f}= \overline{k} \overline{f}$. Then, $\overline{fh} =\overline{fk}$ which implies $fh =fk $ and use the fact that $f$ is a monomorphism to conclude $h=k$ and hence, $\overline{h}= \overline{k}$.

Answer 2

If $w \stackrel g \to x \stackrel f \to y$ in a category $\newcommand{\C}{\textsf{C}} \C$, then $w \stackrel g \leftarrow x \stackrel f \leftarrow y$ in $\C^{\rm op}$. And, if we write $\circ$ for the composition in $\C$ and $\circ’$ for the composition in $\C^{\rm op}$, we have $$ g \circ’\! f = f \circ g. $$ Hence, if $f$ is a mono in $\C$, it is an epi in $\C^{\rm op}$: for any two $w \stackrel {h,k} \leftleftarrows x$ in $\C^{\rm op}$, \begin{align} h \circ’\! f = k \circ’\! f & \iff f \circ h = f \circ k \\ & \implies h=k. \end{align}