In the article "Coherence for compact closed categories", Kelly and Laplaza say
The “category” of small categories and profunctors fails to be a compact closed category only because it fails to be an honest category with associative composition.
What composition are they talking about? What do they mean by an honest category?
There's no consistent way to define the composition, as it involves taking a coend. The traditional way around this is to identify functors between small categories with cocontinuous functors between their free cocompletions, where the composition becomes the standard composition of functors. Unfortunately, the category of locally small cocomplete categories and cocontinuous functors is not closed, as the categories of functors may not be locally small. This can be solved by viewing small categories abd profunctors as a compact closed bicategory.