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Let $A$ be a locally small category and let $\mathbf{Cat}$ be the 2-category of small categories, functors and natural transformations and let $Ps(A)$ be the 2-category of presheaves (the objects are functors, 1-cells are natural transformations and the 2-cells are modifications).

My question is:

Is $Ps(A)$ an accessible category?

I read some things in enriched categories and 2-categories so I think is true but I am not sure.

Somebody can help me with a references?

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  • $\begingroup$ Presheaves of categories? $\endgroup$
    – David Roberts ♦
    Commented Jun 16, 2020 at 4:04
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    $\begingroup$ Your category of presheaves does not need to be legitimate. So, one cannot expect that is is accessible. $\endgroup$
    – Jiří Rosický
    Commented Jun 16, 2020 at 5:16
  • $\begingroup$ You can solve the issue in the comment above by either using small presheaves ncatlab.org/nlab/show/small+presheaf (and I bet that's what you meant when you wrote the definition of $PA$), or by using class-accessible categories sciencedirect.com/science/article/pii/S0022404912000321 $\endgroup$
    – fosco
    Commented Jun 16, 2020 at 8:14

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