It seems that you are using the default settings to define the assembly. With these settings, the contents for the Copy Files task is "** \ bin \ $ (BuildConfiguration) **", while the api web project does not have a configuration configuration folder. Therefore, he cannot find the files for the web api project. To copy these files, add another Copy Files task and configure the settings as follows: 
If you need deployment files for the project, you need to set the assembly definition as follows:
- Add the arguments
/p:DeployOnBuild=true;OutDir="$(build.artifactstagingdirectory)" for the Visual Studio Build step. - Delete the
Copy Files step. - Set the publishing path to
$(build.artifactstagingdirectory)\\_PublishedWebsites for the Publish Build Artifacts step.
Then you should get the deployment files in the drop folder.
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