You have the two bbl, 1-1/16 inch venturi carb, a Holley 2110. They are peculiar to the '56 model only and all had the automatic choke.
Things to check:
Look to see if linkage rod from choke lever that connects to fast idle cam has become disconnected at one end. I believe it is held in place by small wire-clip.
Inspect the condition of the copper heat tube and its connections for looseness or corrosion. The heat tube has a cloth jacket which can be pulled up or down to inspect the tube. Tube is located on driver side just below the automatic choke assembly.
Is the automatic choke adjustment correct? If you have a instruction sheet from a rebuild kit, that will tell you how to adjust choke. Also a 1956 Ford car shop manual has an abundance of rebuild and adjustment instructions for all the carbs offered that year. I highly recommend getting one.