Theorem 1

Let $f$ be a mapping of A into B, let $f'$ be a mapping of B into C, and let $f'' = f' \circ f$. Then :

(a) If $f$ and $f'$ are injections, then $f''$ is an injection. If $r$, $r'$ are retractions of $f$, $f'$, respectively, then $r \circ r'$ is a retraction of $f''$.

(b) If $f$ and $f'$ are surjections, then $f''$ is a surjection. If $s$, $s'$ are sections of $f$, $f'$, respectively, then $s \circ s'$ is a section of $f''$.

(c) If $f''$ is an injection, then $f$ is an injection. If $r''$ is a retraction of $f''$, then $r'' \circ f'$ is a retraction of $f$.

(d) If $f''$ is a surjection, then $f'$ is a surjection. If $s''$ is a section of $f''$, then $f \circ s''$ is a section of $f'$.

(e) If $f''$ is a surjection and $f'$ an injection, then $f$ is a surjection. If $s''$ is a section of $f''$, then $s'' \circ f'$ is a section of $f$.

(f) If $f''$ is an injection and $f$ a surjection, then $f'$ is an injection. If $r''$ is a retraction of $f''$, then $f \circ r''$ is a retraction of $f'$.

For every set E let $\mathrm{I_E}$ denote the identity mapping of E.

(a) We have $r \circ f = \mathrm{I_A}$ and $r' \circ f' = \mathrm{I_B}$, hence

If $f$ and $f'$ are injections, then $f''$ is an injection, by Proposition 8 if $\mathrm{A} \neq \emptyset$, and trivially if $\mathrm{A} = \emptyset$.

(b) We have $f \circ s = \mathrm{I_B}$ and $f' \circ s' = \mathrm{I_C}$, hence

If $f$ and $f'$ are surjections, $f''$ is then a surjection by Proposition 8.

(c) We have $r'' \circ f'' = \mathrm{I_A}$, hence $(r'' \circ f') \circ f = r'' \circ f'' = \mathrm{I_A}$. If $f''$ is an injection, then $f$ is an injection, by Proposition 8 if $\mathrm{A} \neq \emptyset$, and trivially if $\mathrm{A} = \emptyset$.

(d) We have $f'' \circ s'' = \mathrm{I_C}$, hence $f' \circ (f \circ s'') = f'' \circ s'' = \mathrm{I_C}$. If $f''$ is a surjection, then $f'$ is a surjection by Proposition 8.

(e) We have $f'' \circ s'' = \mathrm{I_C}$, and $f'$ is a bijection by (d). Hence

If $f''$ is a surjection and $f'$ an injection, $f$ is then a surjection by Proposition 8.

(f) We have $r'' \circ f'' = \mathrm{I_A}$, and $f$ is a bijection by (c). Hence

If $f''$ is an injection and $f$ a surjection, then $f'$ is an injection, by Proposition 8 if $\mathrm{A} \neq \emptyset$, and trivially if $\mathrm{A} = \emptyset$ (for then we have $\mathrm{B} = f\langle \mathrm{A}\rangle = \emptyset$).