• The First Isomorphism Theorem for Sets. We have an isomorphism of sets[1][2]
    \[ X/\mathord {\sim }_{\mathrm{Ker}\webleft (f\webright )} \cong \mathrm{Im}\webleft (f\webright ). \]

Footnotes

[1] Further Terminology: The set $X/\mathord {\sim }_{\mathrm{Ker}\webleft (f\webright )}$ is often called the coimage of $f$, and denoted by $\mathrm{Coim}\webleft (f\webright )$.
[2] In a sense this is a result relating the monad in $\textbf{Rel}$ induced by $f$ with the comonad in $\textbf{Rel}$ induced by $f$, as the kernel and image
\begin{gather*} \mathrm{Ker}\webleft (f\webright )\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X,\\ \mathrm{Im}\webleft (f\webright )\subset Y \end{gather*}
of $f$ are the underlying functors of (respectively) the induced monad and comonad of the adjunction
of Chapter 6: Constructions With Relations, Item 2 of Proposition 6.3.1.1.2.

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