CatDat

Implication Details

Claim: If a morphism is an epimorphism and is a strong monomorphism, then it is an isomorphism.

Proof: Assume that $m : A \to B$ is a strong monomorphism which is also an epimorphism. Then we apply the orthogonality condition to $$\begin{CD} A @>m>> B \\ @V{\id_A}VV @VV{\id_B}V \\ A @>>m> B \end{CD}$$ to conclude that $m$ is a split epimorphism, and hence an isomorphism.