Fact based communication is a powerful tool.

My take on this is:

If I were you, I would have some tangible/measurable ways in place to assess the quality of the outsourceed works, then communicate to the person in charge.

For example, I might use "on time", "under budget", and "product quality" to document their deliveries, in which "product quality" might be further itemized as

'preject recycled', 'critical', 'medium', or 'minor' issues, etc. and then clearly express your expectations.

Finally, you could use a chart to summarize the trend of their work quality, and emphasize "a certain level of quality is attainable".