Why the compact-open topology of the dual group $G^*$ of a discrete group $G$ coincides with the topology inherited from the product ${\bf T}^G$?

Question

I have following question about a step (tagged in the image below) in the proof of Prop. 2.9.1(b) in Ribes' "Profinite Groups" (p. 60):

$G^*$ has the compact-open topology which is induced by the subbase consisting of $B(K,U) =\{f \in C(G, T) | f(K) \subset U\}$ and $G$ the discrete topology by assumption.

How can $G^*$ be embedded (in topological sense) in $\prod _G T$ so that it's compact-open topology coincides with subgroup topology induced by the product topo of $\prod _G T$?

Here the setting...

... AND the problem...

Answer 1

How can $G^*$ be embedded (in topological sense) in $\prod _G T$ so that it's compact-open topology coincides with subgroup topology induced by the product topo of $\prod _G T$?

In the quoted book is used the natural embedding $G^*\to \bf {T}^G$, $f\mapsto (f(x))_{x\in G}$.