Unit test .net 6 console app using IHostApplicationLifeTime

I have a class (CronClass) that inherits from IHostedService with 2 methods which are StartAsync(CancellationToken) and StopAsync(CancellationToken).

How do you go about unit testing the StartAsync method to verify that the code was executed, using Moq Library?
For example:

public class CronClass: IHostedService
{
    private readonly IHostedApplicationLifetime applicationLifetime;
    private readonly IService service;
    // IHostedApplicationLifetime/IService are injected DI to via the constructor
    public Task StartAsync(CancellationToken cancellationToken)
    {
        applicationLifeTime.ApplicationStarted.Register(() =>
        {
            Task.Run(async () =>
            {
                log.LogInformation("Cron Started");
                await service.Process();
            });
        });
    }
    //...
}

I would start with creating a mock of IHostApplicationLifetime

public class MockHostApplicationLifetime : IHostApplicationLifetime, IDisposable
{
    internal readonly CancellationTokenSource _ctsStart = new CancellationTokenSource();
    internal readonly CancellationTokenSource _ctsStopped = new CancellationTokenSource();
    internal readonly CancellationTokenSource _ctsStopping = new CancellationTokenSource();
    public MockHostApplicationLifetime()
    {
    }
    public void Started()
    {
        _ctsStart.Cancel();
    }
    CancellationToken IHostApplicationLifetime.ApplicationStarted => _ctsStart.Token;
    CancellationToken IHostApplicationLifetime.ApplicationStopping => _ctsStopping.Token;
    CancellationToken IHostApplicationLifetime.ApplicationStopped => _ctsStopped.Token;
    public void Dispose()
    {
        _ctsStopped.Cancel();
        _ctsStart.Dispose();
        _ctsStopped.Dispose();
        _ctsStopping.Dispose();
    }
    public void StopApplication()
    {
        _ctsStopping.Cancel();
    }
}

In your unit test create a mock of IService. Create instance of CronClass and call cronClass.StartAsync. Then start MockHostApplicationLifetime. It will trigger registered callback ApplicationStarted.Register. Then verify that Process() was called.

You are starting the task in Register method, so it can happen that the unit test can finish before the task is created and service.Process is called. In that case I would wait some time before verification.

[Test]
public async Task Test1()
{
    var hal = new MockHostApplicationLifetime();
    var mockService = new Mock<IService>();
    var cronClass = new CronClass(hal, mockService.Object);
    await cronClass.StartAsync(CancellationToken.None);
    hal.Started();
    // maybe not needed, test passed without the delay
    //await Task.Delay(500);
    mockService.Verify(mock => mock.Process());
}