# Factoring Mapping into Surjection and Inclusion

## Theorem

Every mapping $f:S \to T$ can be uniquely factored into a surjection $g$ followed by the inclusion mapping $i_T$.

That is, $f = i_T \circ g$ where:

$g: S \to \Img f: \map g x = \map f x$
$i_T: \Img f \to T: \map {i_T} x = x$

This can be illustrated using a commutative diagram as follows:

$\begin{xy}\[email protected][email protected]+1em { S \[email protected]{-->}[r]^*{g} \ar[rd]_*{f = i_T \circ g} & \Img f \[email protected]{-->}[d]^*{i_T} \\ & T }\end{xy}$

## Proof

From Surjection by Restriction of Codomain, any $f: S \to \Img f$ is a surjection.

The mapping $g: S \to \Img f$ where $\map g x = \map f x$ is therefore also clearly a surjection.

The mapping $g: S \to \Img f: \map g x = \map f x$ is clearly unique, by Equality of Mappings.

From Inclusion Mapping is Injection, $i_T: \Img f \to T$ is an injection.

Likewise, the mapping $i_T: \Img f \to T : \map {i_T} x = x$ is also unique, by its own definition.

$\blacksquare$