Exercise 19, § 2

Let $G, G', G''$ be three topological groups; let $f$ be a strict morphism of $G$ into $G'$, and let $g$ be a strict morphism of $F'$ into $G''$. Show that if $f(G)$ contains the kernel $\overline{g}(e'')$ of $g$ ($e''$ being the identity element of $G''$), then $g \circ f$ is a strict morphism of $G$ into $G''$ (cf. no. 7, Corollary to Proposition 21). Give an example in which $f$ is injective and the preceding condition is not satisfied, and $g \circ f$ is not a strict morphism of $G$ into $G''$ (cf. no. 7, Remark following Proposition 20).