Quarterly seasonal adjustments in official statistics are often not the result of a direct adjustment of the quarterly series, but instead are an indirect adjustment arising from the aggregation of the seasonally adjusted monthly series. However, the temporal aggregation of nonseasonal monthly series to a quarterly frequency can exhibit seasonality; we provide a rigorous framework for understanding how this occurs. To solve the problem, we build on prior work that uses benchmarking to enforce seasonal adjustment adequacy as temporal aggregation is applied, where adequacy is metrized and supplied as a hard constraint to the benchmarking optimization problem. It is vital to use a seasonality diagnostic that examines a time series at high seasonal lags, and can properly capture type I and type II errors, and therefore we propose to utilize new autoregressive seasonality diagnostics in tandem with the proposed benchmarking procedure. We examine the proposed procedure on X-13ARIMA-SEATS seasonal adjustments of several economic time series.