Proposition 1

(1) If $f$ is a surjection of a set E onto a set F, the canonical extension $\hat{f}$ of $f$ is a surjection of $\mathfrak{P}(\mathrm{E})$ onto $\mathfrak{P}(\mathrm{F})$.

(2) If $f$ is an injection of E into F, the canonical extension $\hat{f}$ of $f$ is an injection of $\mathfrak{P}(\mathrm{E})$ into $\mathfrak{P}(\mathrm{F})$.

(1) If $s$ is a section of $f$, then $f \circ s$ is the identity mapping of F, hence $\hat{f} \circ \hat{s}$ is the identity mapping of $\mathfrak{P}(\mathrm{F})$; therefore $\hat{f}$ is surjective and $\hat{s}$ is a section of $\hat{f}$ (§3, no. 8).

(2) The proposition is obvious if $E = \varnothing$, because then $\mathfrak{P}(E) = \{\varnothing\}$. If $E \ne \varnothing$ and if $r$ is a retraction of $f$, then $r \circ f$ is the identity mapping of $E$, so that $\hat{r} \circ \hat{f}$ is the identity mapping of $\mathfrak{P}(E)$; therefore $\hat{f}$ is injective, and $\hat{r}$ is a retraction of $\hat{f}$ (§3, no. 8).