The following elements should be considered in addressing this question:

1. The definition of assets under management (AUM) is provided in Article 4(1)(27) of Regulation (EU) 2019/2033 (IFR). 
2. AUM is defined as “[…] the value of assets that an investment firm manages for its clients under both discretionary portfolio management and nondiscretionary arrangements constituting investment advice of an ongoing nature”.
3. ‘Investment advice of ongoing nature’ is defined in Article 4(1)(21) of the IFR as “the recurring provision of investment advice as well as the continuous or periodic assessment and monitoring or review of a client portfolio of financial instruments, including of the investments undertaken by the client on the basis of a contractual arrangement”. Therefore, the ‘monitoring’ activity is explicitly mentioned in this definition.
4. The notion of “monitoring” is not further specified in the IFR or in the MiFID. As specified in recital 24 of the IFR, “K‐AUM captures the risk of harm to clients from an incorrect discretionary management of client portfolios or poor execution and provides reassurance and client benefits in terms of the continuity of service of ongoing portfolio management and investment advice”.
5. Monitoring activities that entail advice on the portfolios, whether the investment firm is authorised to act on behalf of the client or not, may include, but often not limited to: analysis on the performance (e.g., comparison with benchmarks and alternative scenarios analysis), asset allocation or concierge services. 

Against this background:

1. Assets related to monitoring activities that involve the mere reception and aggregation of statements on the value of assets under management by other undertakings should not be included in the AUM calculation.
2. Conversely, monitoring activities that entail advice on the portfolios, whether the investment firm is authorised to act on behalf of the client or not, should be included in the AUM calculation.
3. If the investment firm is not able to discriminate between the part of the assets that are related to the mere reception and aggregation of statements and the assets where the monitoring activities entail advice on the portfolios, then all client’s assets should be included in the AUM calculation.

An investment firm may provide both ‘investment advice’ and ‘monitoring’ activities on a client’s asset, incurring in a single ‘asset under management’ (AUM) capital requirement, i.e., the K-factor is calculated only once.

However, when these activities are performed by different investment firms on the same assets of a single client, all investment firms involved should calculate the AUM on those assets. This would be the case if, for example, the assets are subject to advisory activities by an investment firm and monitored by a different investment firm. This case should not be considered as ‘double counting’ because, in case of monitoring of the client’s portfolios, the potential risk is a misrepresentation or an inaccurate aggregation of the portfolios, which is different from the risks that arise from providing portfolio management or ongoing advice.