Did any one solve this problem??

...not working on a MBP A1278 (Catalina)

The older "A" numbers were not unique to a year. You have to do "About this Mac.. to see your sub-model year that helps most here.

"A1278" was used on five different 13-inch Unibody Macbook Pros from Mid 2009 through Mid 2012. Within those, there are different macOS version imitations:

• Mid 2009: max of OS10.11 "El Captian"
• Mid 2010-Late 2011: max of OS 10.13
• Mid 2012: max of OS 10.15 "Catalina."

Please post which you have. Thanks!

A1316 is Apple Cinema Display LED (27-Inch)

Have you POSITIVELY identified is as an A1316:

a convenient single cable with three connectors (Mini DisplayPort, MagSafe, and USB 2.0)

and not its even twin:

Apple Cinema Display LED (27-Inch) A1407 ???

if you are certain that is what you have the connection is simple.

if this is your computer:

Apple MacBook "Core 2 Duo" 2.4 13" (Unibody) Specs

Identifiers: Late 2008 A1278

that computer does NOT run Catalina.

Then you connect its ThunderBolt/MiniDisplayPort output to the Apple display's MiniDisplayPort input with a Mini DisplayPort cable.

although it can be exact, using only the model numbers sends readers scurrying to look them up by model number, rather than common names. There is no additional information such as model-year for cross checking.

What is the problem?