**Core Concept**
The lac operon is a genetic regulatory system in E. coli that controls the expression of genes involved in lactose metabolism. The lac operon is a classic example of gene regulation through the interaction of transcription factors and DNA. Catabolite gene activator protein (CAP), also known as cAMP receptor protein (CRP), plays a crucial role in the regulation of the lac operon.

**Why the Correct Answer is Right**
CAP binds to the lac operator region in the presence of cyclic AMP (cAMP) and promotes the binding of RNA polymerase to the promoter region, thereby activating the transcription of the lac genes. CAP binds to a specific DNA sequence near the lac operator region, known as the CAP binding site, and forms a complex with cAMP. This complex then interacts with RNA polymerase to facilitate the initiation of transcription. The binding of CAP to the lac operator region is essential for the activation of the lac genes in response to the presence of lactose.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because CAP is not responsible for repressing the lac genes. In fact, CAP plays a crucial role in activating the lac genes in response to the presence of lactose.
* **Option B:** This option is incorrect because CAP does not directly bind to the lac repressor protein. Instead, CAP binds to a specific DNA sequence near the lac operator region and promotes the binding of RNA polymerase to the promoter region.
* **Option D:** This option is incorrect because CAP is not responsible for the synthesis of lactose. CAP plays a role in regulating the expression of genes involved in lactose metabolism, but it does not directly participate in the synthesis of lactose.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that CAP is a global regulator of gene expression in E. coli, and its binding to specific DNA sequences can have far-reaching consequences for gene regulation. CAP's ability to bind to cAMP and activate gene expression is a key mechanism by which E. coli adapts to changes in its environment.

**Correct Answer: C. Promotes the binding of RNA polymerase to the promoter region.**