Proposition
(limits of presheaves are computed objectwise)
Let be a category and write for its category of presheaves. Let moreover be a small category and consider any functor
hence a -shaped diagram in the category of presheaves.
Then
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The limit of exists, and is the presheaf which over any object is given by the limit in Set of the values of the presheaves at :
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The colimit of exists, and is the presheaf which over any object is given by the colimit in Set of the values of the presheaves at :
Proof
This is elementary, but we spell it out in detail.
We discuss the case of limits, the other case is formally dual.
Observe that there is a canonical equivalence
where is the product category.
This makes manifest that a functor is equivalently a diagram of the form
Then observe that taking the limit of each โhorizontal rowโ in such a diagram indead does yield a presheaf on , in that the construction extends from objects to morphisms, and uniquely so: This is because for any morphism in , a cone over induces a cone over , by vertical composition with
From this, the universal property of limits of sets implies that there is a unique morphism between the pointwise limits which constitutes a presheaf over
and that is the tip of a cone over the diagram in presheaves.
Hence it remains to see that this cone of presheaves is indeed universal.
Now if is any other cone over in the category of presheaves, then by the universal property of the pointswise limits, there is for each a unique morphism of cones in sets
Hence there is at most one morphisms of cones of presheaves, namely if these components make all their naturality squares commute.
But since everything else commutes, the two ways of going around this diagram constitute two morphisms from a cone over to the limit cone over , and hence they must be equal, by the universal property of limits.