11.4.1 Isomorphisms
Let $\mathcal{C}$ be a category.
A morphism $f\colon A\to B$ of $\mathcal{C}$ is an isomorphism if there exists a morphism $\smash {f^{-1}\colon B\to A}$ of $\mathcal{C}$ such that
\begin{align*} f\circ f^{-1} & = \operatorname {\mathrm{id}}_{B},\\ f^{-1}\circ f & = \operatorname {\mathrm{id}}_{A}. \end{align*}
We write $\operatorname {\mathrm{Iso}}_{\mathcal{C}}(A,B)$ for the set of all isomorphisms in $\mathcal{C}$ from $A$ to $B$.
Let $\mathcal{C}$ be a category.
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1.
Two-Out-of-Three. Let
be a diagram in $\mathcal{C}$. If two out of the three morphisms among $f$, $g$, and $g\circ f$ are isomorphisms, then so is the third.
Omitted.
