Closure operator and set of fixpoint

Question

In chapter 2.2 of

Giacobazzi, Roberto; Ranzato, Francesco, Uniform closures: Order-theoretically reconstructing logic program semantics and abstract domain refinements, Inf. Comput. 145, No.2, 153-190 (1998). ZBL0921.68057.

it's said:

An (upper) closure operator (or simply closure) on a poset $C$ is an operator $\rho:C \to C$ monotone, idempotent and extensive (i.e., $\forall x \in C . x \le \rho(x)$). We denote by $uco(C)$ the set of all closure operators on the poset $C$. If $C$ is a complete lattice then each closure operator $uco(C)$ is uniquely determined by the set of its fixpoints, which is its image $\rho(C)$

Where can I find a proof of the phrase in bold?