Isomorphism (category theory)

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Definition

Given a pair of arrows, $A\mathop{\longrightarrow}^fB$ and $B\mathop{\longrightarrow}^g A$ in a category $\mathscr{C}$, we say that $f$ and $g$ are isomorphisms if:

• $g\circ f=\text{Id}_A$ and $f\circ g=\text{Id}_B$

We may also call $f$ and $g$ an inverse pair of isomorphisms. For clarity I say again: both $f$ and $g$ are themselves isomorphisms

See also

• Types of category arrows
• Bimorphism