Let $A$ and $B$ be sets, and let $f: A \to B$ be a function. Define the image of $f$ by
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\begin{align*}
\operatorname{im} f := \{ b \in B : \text{there exists } a \in A \text{ such that } f(a)=b\}.
\end{align*}
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Define $\tilde f: A \to \operatorname{im} f$ by $\tilde f(a)=f(a)$ for every $a \in A$, and let $\iota: \operatorname{im} f \to B$ be the inclusion map, so $\iota(y)=y$ for every $y \in \operatorname{im} f$. Then $\tilde f$ is surjective and
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\begin{align*}
f = \iota \circ \tilde f.
\end{align*}