Limits of monoid objects. #

If C has limits, so does Mon_ C, and the forgetful functor preserves these limits.

(This could potentially replace many individual constructions for concrete categories, in particular MonCat, SemiRingCat, RingCat, and AlgebraCat R.)

We construct the (candidate) limit of a functor F : J ⥤ Mon_ C by interpreting it as a functor Mon_ (J ⥤ C), and noting that taking limits is a lax monoidal functor, and hence sends monoid objects to monoid objects.

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    The image of the proposed limit cone for F : J ⥤ Mon_ C under the forgetful functor forget C : Mon_ C ⥤ C is isomorphic to the limit cone of F ⋙ forget C.

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