Proposition 2

(1) If $u$ is a surjection of $E'$ onto $E$ and $v$ an injection of $F$ into $F'$, then the mapping $f \mapsto v \circ f \circ u$ is injective.

(2) If $u$ is an injection of $E'$ into $E$ and $v$ a surjection of $F$ onto $F'$, then the mapping $f \mapsto v \circ f \circ u$ is surjective.

We shall assume that the sets $E$, $E'$, $F$, $F'$ are all non-empty, since otherwise the proposition is trivially verified.

(1) Let $s$ be a section of $u$ and let $r$ be a retraction of $v$ (§3, Definition 11). Then $r \circ (v \circ f \circ u) \circ s = I_F \circ f \circ I_E = f$, so that

is injective.

(2) Let $r'$ be a retraction of $u$ and let $s'$ be a section of $v$. For every mapping $f' : E' \to F'$ we have $v \circ (s' \circ f' \circ r') \circ u = f'$, which shows that the mapping $f \mapsto v \circ f \circ u$ is surjective.