Traditional remote data possession auditing mechanisms are usually divided into two types: private auditing and public auditing. Informally, private auditing only allows the original data owner to check the integrity of its outsourced data, whereas anyone is allowed to perform the checking task in public auditing. However, in many practical applications, the data owner expects some designated verifier (instead of others) to audit its data file, which cannot be covered by the existing private or public auditing protocols. Thus, in a recent work, Yan et al. proposed a new auditing technique with a designated verifier [IEEE Systems Journal, 12(4): 1788-1797, 2020]. Nevertheless, we note that this protocol suffers from complicated management of certificates and hence heavily relies on public key infrastructure. To overcome this shortcoming, in this paper, we propose an identity-based auditing protocol with a designated verifier, which not only avoids the introduction of certificates, but also has the desired property of only allowing specific verifier to audit. Its security is based on the classical computational Diffie-Hellman and Weil Diffie-Hellman assumptions. Finally, performance analysis shows that our proposed protocol is very efficient and suitable for some real-life applications.