Fix a small category I and a diagram D ∶ I → C of shape I. Then the limit of D, if it exists, consists of an object l ∈ C and a natural isomorphism
Cone ∶ HomC(−,l) ≅ HomCI((−)I,D) ∶ Uni
in the category SetCop.
This is intuitively plausible if we recall the definition of limits. Recall that a cone under D consists of an object l ∈ C and a natural transformation Λ ∶ lI ⇒ D. We say that the cone (l,Λ) is the the limit of D if, for any other cone Φ ∶ cI ⇒ D, there exists a unique arrow υ ∶ c → l making the following diagram in CI commute:
The map sending the cone Φ ∶ cI ⇒ D to the unique arrow υ ∶ c → l is the desired function HomCI (cI,D) → HomC(c,l). Furthermore, it’s clear that this function is a bijection since we can pull back any arrow α ∶ c → l to the cone Λ ○ αI ∶ cI ⇒ D. The main difficulty is to show that the data of naturality for these bijections is equivalent to the data of the canonical cone Λ ∶ lI ⇒ D.
Proof: First assume that the limit of D exists and is given by the cone (limID,Λ). In this case we want to define a family of bijections
Unic ∶HomcI(cI,D) →~ Hom (c,limID)
that is natural in c ∈ Cop. (Then the inverse Cone ∶= Uni−1 is automatically natural. So consider any element Φ ∈ HomCI(cI,D), i.e., any cone Φ ∶ cI ⇒ D. By the definition of limits we know that there exists a unique arrow υ ∶ c → limID making the following diagram commute:
Therefore the assignment Unic(Φ) ∶= υ defines an injective function (recall that the functor (−)I is faithful, so that υ1I = υ2I implies υ1 = υ2). To see that Unic is surjective, consider any arrow α ∶ c → limID in C. We want to define a cone Φα ∶ cI ⇒ D with the property that Unic(Φα) = α. By definition of Unic this means that we must have Φα ∶= Λ ○ αI — in other words, we must have (Φα)i ∶= Λi ○ α ∀ indices i ∈ I. And note that this does define a natural transformation Φα ∶ cI ⇒ D since for all arrows δ ∶ i ∈ j in I we have
D(δ) ○ (Φα)i =D(δ) ○ (Λi ○ α)
= (D(δ) ○ Λi) ○ α
= Λj ○ α (Naturality of Λ)
We conclude that Unic is a bijection. To see that Unic is natural in c ∈ Cop, consider any arrow γ ∶ c1 → c2 in C (i.e., any arrow γ ∶ c2 → c1 in C). We want to show that the following diagram commutes:
And to see this, consider any cone Φ ∶ cI1 ⇒ D. By composing with the natural transformation γI ∶ cI2 ⇒ cI1 we obtain the following commutative diagram in CI:
Since the diagonal embedding (−)I ∶ C → CI is a functor, the bottom arrow is given by
(Unic1 (Φ))I ○ γI = (Unic1 (Φ) ○ γ)I
But by the definition of the function Unic2 this arrow also equals (Unic2(Φ ○ γI))I
Then since (−)I is a faithful functor we conclude that
Unic2 (Φ ○ γI) = Unic1 (Φ) ○ γ
and hence the desired square commutes. Conversely, consider an object l ∈ C and suppose that we have a bijection
Conec ∶HomC(c,l) ←→ HomCI(cI,D) ∶ Unic
that is natural in c ∈ Cop. In other words, suppose that for each arrow γ ∶ c1 → c2 in Cop (i.e., for each arrow γ ∶ c2 → c1 in C) we have a commutative square:
We want to show that this determines a unique cone Λ ∶ lI ⇒ D such that (l, Λ) is the limit of D. The only possible choice is to define Λ ∶= Conel(idl). Now given any cone Φ ∶ cI ⇒ D we want to show that there exists a unique arrow υ ∶ c → l with the property Λ ○ υI = Φ.
So suppose that there exists some arrow υ ∶ c → l with the property Λ ○ υI = Φ. By substituting γ ∶= υ into the above diagram we obtain a commutative square:
Then following the arrow idl ∈ HomC(l, l) around the square in two different ways gives
idl ○ υ = Unic(Conel(idl) ○ υI)
υ = Unic(Λ ○ υI)
υ = Unic(Φ)
Thus there exists at most one such arrow υ. To show that there exists at least one such arrow, we must check that the arrow Unic(Φ) actually does satisfy Λ ○ (Unic(Φ))I = Φ. Indeed, by substituting υ ∶= Unic(Φ) into the above diagram we obtain a commutative square:
Then following the arrow idl ∈ HomC(l,l) around the
Conel(idl) ○ (Unic(Φ))I) = Conec(idl ○ Unic(Φ)) Λ ○ (Unic(Φ))I
Λ ○ (Unic(Φ))I = Φ
square in two ways gives as desired.
[Remark: We have proved that the limit of a diagram D ∶ I → C, if it exists, consists of an object limID ∈ C and a natural isomorphism
HomC(−, limID) ≅ HomCI((−)I,D) of functors Cop → Set. It turns out that if all limits of shape I exist in C then there is a unique way to extend this to a natural isomorphism
HomC(−,limI−) ≅ HomCI((−)I,−)
of functors Cop × CI → Set, and hence that we have an adjunction (−)I ∶ C ⇄ CI ∶ limI. However, we don’t need this result right now so we won’t prove it. Dually, the colimit of D, if it exists, consists of an object colimID ∈ C and a natural isomorphism HomC(colimID, −) ≅ HomCI(D, (−)I) of functors C → Set. If all colimits of shape I exist in C then this extends uniquely to an adjunction colimI ∶ CI ⇄ C ∶ (−)I. This explains the title of the previous lemma.]
Let L ∶ C ⇄ D ∶ R be an adjunction and consider a diagram D ∶ I → D of shape I in D. If the diagram D ∶ I → D has a limit cone Λ ∶ lI ⇒ D then the composite diagram RI(D) ∶ I → C also has a limit cone, which is given by RI(Λ) ∶ R(l)I ⇒ RI(D).
In this proof we will write limID ∶= l ∈ D, and we will just assume that the limit object limI RI(D) ∈ C exists. Now we want to show that the following objects are isomorphic in C : R(limID) ≅ limI RI (D). (We will ignore the data of the limit cone.)
So assume that L ∶ C ⇄ D ∶ R is an adjunction. Then we have the following sequence of bijections, each of which is natural in c ∈ Cop:
Homc(c, R(limID)) →~ Homc(L(c), limID) (L ⊣ R)
→~ HomDI(L(c),D) (Diagonal ⊣ Limit)
→~ HomDI (LI (cI),D)
→~ Homc(c,limIRI(D)) (Diagonal ⊣ Limit)
By composing these we obtain a family of bijections
Homc(c,R(limID)) →~ Homc(c,limI RI(D))
that is natural in c ∈ Cop. In other words, we obtain an isomorphism of hom functors HR(limI(D)) ≅ HlimI RI(D) in the category SetCop. Then since the Yoneda embedding H(−) : C → SetCop is essentially injective (from the Embedding Lemma), we obtain an isomorphism of objects R(limID) ≅ limI RI(D) in the category C.