Extending a Map by One Point, and Extending a Bijection by One Point

In the setting of Sets and Maps: Ordinary Notation, let $A$ and $B$ be sets, let $g:A\to B$ be a map, let $p$ be a set with $p\notin A$, and let $c\in B$; $\{p\}$ and $\{c\}$ are singletons.

(i) There is exactly one map $h:A\cup\{p\}\to B$ such that $h(u)=g(u)$ for every $u\in A$ and $h(p)=c$.

(ii) Let moreover $C$ be a subset of $B$ such that $g:A\to C$ is a bijection, and let $c\notin C$. Then the map $h$ of (i) is a bijection $h:A\cup\{p\}\to C\cup\{c\}$.