Given categories and , the functor category – written or – is the category whose
-accessible functors from a -accessible category to any locally -presentable category form a locally -presentable category.
Cocontinuous functors between locally presentable categories form a locally presentable category. More precisely, if and are locally -presentable, then so is .
Continuous accessible functors between locally presentable categories form the opposite of a locally presentable category. More precisely, if and are locally -presentable, then so is .
Indeed, the point is this: given a -accessible category ( essentially small), the category of -accessible functors (for arbitrary ; here by “-accessible” we mean simply “preserves -filtered colimits”) is naturally equivalent to the category of all . It should be well known that:
If is accessible, then so is .
If is locally -presentable, then so is .
Colimit-preserving functors out of a locally -presentable category are -accessible.
A right adjoint between locally -presentable categories is -accessible if and only if its left adjoint is strongly -accessible (i.e. preserves -presentable objects as well as -filtered colimits); and every limit-preserving accessible functor between locally presentable categories is a right adjoint.
Statements 1 and 2 are proved in [Adamek and Rosický, Locally presentable and accessible categories], statement 3 is obvious, and statement 4 is a straightforward exercise. Thus the claims follow.
In general, accessible functors between accessible categories do not form an accessible category due to size issues. The best one can hope for is a class-accessible category. Let be an accessible category that is not essentially small. Consider the category of all accessible functors . This is the same as the smallest full replete subcategory of containing all representable functors and closed under small colimits. In particular, is accessible if and only if locally presentable. We claim is not accessible.
Indeed, suppose has a small generating family, say . Then for some regular cardinal , every member of is -accessible. So consider for some object that is not -presentable. (Such an exists because is not essentially small.) Since generates, there is a small diagram of -accessible functors whose colimit is . But then is a retract of a -accessible functor and hence -accessible: a contradiction. That said, is a class-locally presentable category.
If and are small, then is also small.
If is small and is locally small, then is still locally small.
Even if and are locally small, if is not small, then will usually not be locally small.