Step 1: Understanding the symptoms.
The patient has a functional arteriovenous (AV) fistula in his left arm for dialysis, and his complaint of coldness in the left hand is likely due to a vascular issue related to the AV fistula. The condition he is most likely suffering from is steal syndrome. Steal syndrome occurs when there is an excessive diversion of blood from the distal vascular bed (in this case, the hand) due to the increased flow through the AV fistula, leading to ischemia or a sensation of coldness and discomfort in the affected limb.
Step 2: Conclusion.
The patient is most likely suffering from dialysis access-related steal syndrome due to the arteriovenous fistula.