Mas-Colell, Whinston and Green in Microeconomic Theory describe the direct revelation mechanism as it follows:
The employed notation is the following:
$θ_i$: Player i's type
$Θ_i$: Set of types for player i
$S_i$ Set of strategies for player i
f: Social choice function
g: Mechanism outcome function
X: Set of collective choices
If $S_i$ is a set of strategies and $Θ_i$ is a set of types, how can $S_i = Θ_i$ ?