David Lidington replaces Damian Green as Cabinet Office minister

This means he will take on the Westminster coordinating role (chairing committees etc) that Damian Green had.

But May has not made Lidington first secretary of state, the title that Green also had.

The Minister for the Cabinet Office is in overall charge of and responsible for the policy and work of the department, and attends Cabinet. This position is often held with another, in this instance it's the Duchy of Lancaster.

The Chancellor of the Duchy of Lancaster is, in modern times, a sinecure office in the government of the United Kingdom.

A sinecure (from Latin sine = "without" and cura = "care") means an office that requires or involves little or no responsibility, labour, or active service.