Add mono+epi = iso for abelian cats

Thanks to Xueping Huang
https://stacks.math.columbia.edu/tag/00ZX#comment-8377

CONTRIBUTORS

@@ -208,6 +208,7 @@ Tim Holzschuh
 Ray Hoobler
 John Hosack
 Xiaowen Hu
+Xueping Huang
 Yuhao Huang
 Yu-Liang Huang
 Logan Hyslop

homology.tex

@@ -419,7 +419,7 @@ \section{Karoubian categories}
 \Mor_\mathcal{D}(W, X)
 )
 $$
-if representable if and only if $e$ has a kernel. Note that for any
+is representable if and only if $e$ has a kernel. Note that for any
 abelian group $A$ and idempotent endomorphism $e : A \to A$ we have
 $$
 \Ker(e : A \to A)
@@ -510,8 +510,9 @@ \section{Abelian categories}
 \label{lemma-characterize-injective}
 Let $f : x \to y$ be a morphism in an abelian category $\mathcal{A}$. Then
 \begin{enumerate}
-\item $f$ is injective if and only if $f$ is a monomorphism, and
-\item $f$ is surjective if and only if $f$ is an epimorphism.
+\item $f$ is injective if and only if $f$ is a monomorphism,
+\item $f$ is surjective if and only if $f$ is an epimorphism, and
+\item $f$ is an isomorphism if and only if $f$ is injective and surjective.
 \end{enumerate}
 \end{lemma}
 
@@ -524,6 +525,10 @@ \section{Abelian categories}
 $\Mor_\mathcal{A}(z, x) \to \Mor_\mathcal{A}(z, y)$
 is injective for all $z$ if and only if $f$ is a monomorphism.
 The proof of (2) is similar.
+For the proof of (3) note that an isomorphism is both a monomorphism
+and epimorphism, which by (1), (2) proves $f$ is injective
+and surjective. If $f$ is both injective and surjective, then
+$x = \Coim(f)$ and $y = \Im(f)$ whence $f$ is an isomorphism.
 \end{proof}
 
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