It is common in construction and other agreements (written or oral) for one party to “assume” the liability of another. This is sometimes referred to as a “hold harmless” agreement. The extent to which one holds another harmless varies from contract to contract, job to job, etc.
To assume the liability of another, regardless of extent, is a voluntary undertaking which increases your exposure to loss. A standard Commercial General Liability policy does cover this additional exposure subject to certain exclusions.