The opposite of a category $C$, written $C^{op}$, has the same objects and arrows as $C$, but the arrows are reversed in direction (domains become codomains, and viceversa):

$f: A \to B$ in $C^{op}$ $\Leftrightarrow$ $f: B \to A$ in $C$.

Composition, $f \circ g$ in $C^{op}$ is defined as the arrow $g \circ f$ in $C$ (which remember, points in the opposite direction. Direction is defined with the $dom, cod$ maps, so they are separate from the arrows themselves..)

This allows us to describe a duality principle:

If we have an statement $\phi$, which holds in $C$. Then, the same statement, but with arrows reversed (and compositions flipped) $\phi^*$ holds in $C^{op}$.

In particular $C^{op}$ is a category, because the axioms of a category are self-dual.

In this sense, monic and epic are dual.

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