Describe construction of finite automata that accepts the language Double-Letter(L)
The idea is to start with an automaton for $L$, and replace each transition labeled $\sigma$ by a path of length 2 both of whose edges are labeled $\sigma$. In order to make it a DFA, we also add a sink state. (This might be unnecessary in your definition of DFA.)

In more detail, suppose we start with a DFA $(Q,q_0,\delta,F)$ for $L$ (here $Q$ is the set of states, $q_0$ is the initial state, $\delta$ is the transition function, and $F$ is the accepting states). We construct a DFA whose set of states is $Q \cup Q \times \Sigma \cup \{q_{\mathit{sink}}\}$, where $\Sigma$ is the alphabet. The initial state is still $q_0$, and the accepting states are still $F$. The transition function $\delta'$ is defined as follows:


$\delta'(q,\sigma) = (q,\sigma)$ (this is a member of $Q \times \Sigma$).
$\delta'((q,\sigma),\sigma) = \delta(q,\sigma)$.
$\delta'((q,\sigma),\tau) = q_{\mathit{sink}}$ if $\tau \neq \sigma$.
$\delta'(q_{\mathit{sink}},\sigma) = q_{\mathit{sink}}$ for all $\sigma \in \Sigma$.


A simpler way of proving that the regular languages are closed under Double-Letter is using regular expressions: given a regular expression for $L$, simply double each letter. This is a special case of closure under homomorphism and regular substitution.