Remark [efr-50T5]

Suppose \mathcal {M}, \mathcal {C}, \mathcal {D} are small categories. Then the coend defining \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D})\left ( {A \choose X}, {B \choose Y} \right ) is a small colimit of small sets, hence again small. Since clearly the set of objects \operatorname {\mathbf {ob}} \mathcal {C} \times \operatorname {\mathbf {ob}} \mathcal {D} is small, \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D}) is again a small category.

However, if \mathcal {M},\mathcal {D},\mathcal {C} are merely assumed to be locally small, we can not guarantee the same is true of \mathsf {Optic}_\mathcal {M}(\mathcal {C},\mathcal {D}), since the hom-sets are now defined by a coend/colimit with large indexing category. However, in many special cases, it can still be seen to be locally small, such as in the Cartesian case (where \mathsf {Lens}(\mathcal {C}) is clearly locally small).

In this thesis, we will not delve further into this subtlety, simply working inside some universe where all our categories are small.