Based on prior knowledge, we can analyze the relationship between the feature "month" and the task of whether the client subscribes to a term deposit.

From the provided categories of the feature "month" ('may', 'jun', 'jul', 'aug', 'oct', 'nov', 'dec', 'jan', 'fe', 'mar', 'apr', 'sep'), it appears to represent the last contact month of the year.

To analyze the relationship, we can examine the frequency or distribution of each month category for both the 'yes' and 'no' classes of the target variable.

Here is the dictionary with the relevant analysis:

```json
{
  "no": ["may", "jun", "jul", "aug", "oct", "nov", "dec"],
  "yes": ["mar", "apr", "sep"]
}
```

According to the analysis, the months of ['may', 'jun', 'jul', 'aug', 'oct', 'nov', 'dec'] are more likely to have clients who did not subscribe to a term deposit ('no'). On the other hand, the months of ['mar', 'apr', 'sep'] are more likely to have clients who did subscribe to a term deposit ('yes').

Note: The months 'jan' and 'fe' are excluded from the dictionary as it is mentioned in the task requirements that it is not necessary to include hard-to-predict month values.