Fix a small category * I* and a diagram

*of shape*

**D ∶ I → C***. Then the limit of*

**I***, if it exists, consists of an object*

**D***and a natural isomorphism*

**l ∈ C****Cone ∶ Hom _{C}(−,l) ≅ Hom_{C}I((−)^{I},D) ∶ Uni**

in the category * Set^{Cop}*.

This is intuitively plausible if we recall the definition of limits. Recall that a cone under * D* consists of an object

*and a natural transformation*

**l ∈ C***. We say that the cone*

**Λ ∶ lI ⇒ D***is the the limit of*

**(l,Λ)***if, for any other cone*

**D***, there exists a unique arrow*

**Φ ∶ c**^{I}⇒ D*making the following diagram in*

**υ ∶ c → l***commute:*

**C**^{I}The map sending the cone * Φ ∶ c^{I} ⇒ D* to the unique arrow

*is the desired function*

**υ ∶ c → l***. Furthermore, it’s clear that this function is a bijection since we can pull back any arrow*

**Hom**_{CI}(c^{I},D) → Hom_{C}(c,l)*to the cone*

**α ∶ c → l***. The main difficulty is to show that the data of naturality for these bijections is equivalent to the data of the canonical cone*

**Λ ○ α**^{I}∶ c^{I}⇒ D*.*

**Λ ∶ l**^{I}⇒ D* Proof*: First assume that the limit of

*exists and is given by the*

**D***. In this case we want to define a family of bijections*

**cone (lim**_{I}D,Λ)**Uni _{c} ∶Hom_{cI}(c^{I},D) →^{~} Hom (c,lim_{I}D)**

that is natural in * c ∈ C^{op}*. (Then the inverse

*is automatically natural. So consider any element*

**Cone ∶= Uni**^{−1}*, i.e., any cone*

**Φ ∈ Hom**_{CI}(c^{I},D)*. By the definition of limits we know that there exists a unique arrow*

**Φ ∶ c**^{I}⇒ D*making the following diagram commute:*

**υ ∶ c → lim**_{I}DTherefore the assignment * Uni_{c}(Φ) ∶= υ* defines an injective function (recall that the functor

*is faithful, so that*

**(−)**^{I}*implies*

**υ**_{1}^{I}= υ_{2}^{I}*). To see that*

**υ**_{1}= υ_{2}*is surjective, consider any arrow*

**Uni**_{c}*. We want to define a cone*

**α ∶ c → lim**_{I}D in C*with the property that*

**Φ**_{α}∶ c^{I}⇒ D*. By definition of*

**Uni**_{c}(Φ_{α}) = α*this means that we must have*

**Uni**_{c}*— in other words, we must have*

**Φ**_{α}∶= Λ ○ α^{I}*indices*

**(Φ**_{α})_{i}∶= Λ_{i}○ α**∀***. And note that this does define a natural transformation*

**i ∈ I***since for all arrows*

**Φ**_{α}∶ c^{I}⇒ D*we have*

**δ ∶ i ∈ j in I****D(δ) ○ (Φ _{α})_{i} =D(δ) ○ (Λ_{i} ○ α)**

**= (D(δ) ○ Λ _{i}) ○ α**

**= Λ _{j} ○ α (Naturality of Λ)**

**= (Φ _{α})_{j}**

We conclude that * Uni_{c}* is a bijection. To see that Uni

_{c}is natural in

*, consider any arrow*

**c ∈ C**^{op}*(i.e., any arrow*

**γ ∶ c**_{1}→ c_{2}in C*). We want to show that the following diagram commutes:*

**γ ∶ c**_{2}→ c_{1}in CAnd to see this, consider any * cone Φ ∶ c^{I}_{1} ⇒ D*. By composing with the natural transformation

*we obtain the following commutative diagram in*

**γ**^{I}∶ c^{I}_{2}⇒ c^{I}_{1}*:*

**C**^{I}Since the diagonal embedding * (−)^{I} ∶ C → C^{I}* is a functor, the bottom arrow is given by

**(Uni _{c1} (Φ))^{I} ○ γ^{I} = (Uni_{c1} (Φ) ○ γ)^{I}**

But by the definition of the function * Uni_{c2}* this arrow also equals

**(Uni**_{c2}(Φ ○ γ^{I}))^{I}Then since (−)^{I} is a faithful functor we conclude that

**Uni _{c2} (Φ ○ γ^{I}) = Uni_{c1} (Φ) ○ γ**

and hence the desired square commutes. Conversely, consider an object * l ∈ C* and suppose that we have a bijection

**Cone _{c} ∶Hom_{C}(c,l) ←→ Hom_{CI}(c^{I},D) ∶ Uni_{c}**

that is natural in * c ∈ C^{op}*. In other words, suppose that for each arrow

*(i.e., for each arrow*

**γ ∶ c**_{1}→ c_{2}in C^{op}*) we have a commutative square:*

**γ ∶ c**_{2}→ c_{1}in CWe want to show that this determines a unique * cone Λ ∶ l^{I} ⇒ D* such that

*is the limit of*

**(l, Λ)***. The only possible choice is to define*

**D***. Now given any cone*

**Λ ∶= Cone**_{l}(id_{l})*we want to show that there exists a unique arrow*

**Φ ∶ c**^{I}⇒ D*with the property*

**υ ∶ c → l***.*

**Λ ○ υ**^{I}= ΦSo suppose that there exists some arrow * υ ∶ c → l* with the property

*. By substituting*

**Λ ○ υ**^{I}= Φ*into the above diagram we obtain a commutative square:*

**γ ∶= υ**Then following the arrow * id_{l} ∈ Hom_{C}(l, l)* around the square in two different ways gives

**id _{l} ○ υ = Uni_{c}(Cone_{l}(id_{l}) ○ υ^{I})**

**υ = Uni _{c}(Λ ○ υ^{I})**

**υ = Uni _{c}(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 * Uni_{c}(Φ)* actually does satisfy

*. Indeed, by substituting*

**Λ ○ (Uni**_{c}(Φ))^{I}= Φ*into the above diagram we obtain a commutative square:*

**υ ∶= Uni**_{c}(Φ)Then following the arrow * id_{l} ∈ Hom_{C}(l,l)* around the

**Cone _{l}(id_{l}) ○ (Uni_{c}(Φ))^{I}) = Cone_{c}(id_{l} ○ Uni_{c}(Φ)) Λ ○ (Uni_{c}(Φ))^{I}**

**= Cone _{c}(Uni_{c}(Φ))**

**Λ ○ (Uni _{c}(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

*and a natural isomorphism*

**lim**_{I}D ∈ C* Hom_{C}(−, lim_{I}D) ≅ Hom_{CI}((−)^{I},D)* of functors

*. It turns out that if all limits of shape*

**C**^{op}→ Set*exist in*

**I***then there is a unique way to extend this to a natural isomorphism*

**C****Hom _{C}(−,lim_{I}−) ≅ Hom_{CI}((−)^{I},−)**

of functors * C^{op} × C^{I} → Set*, and hence that we have an adjunction

*. However, we don’t need this result right now so we won’t prove it. Dually, the colimit of*

**(−)**^{I}∶ C ⇄ C^{I}∶ lim_{I}*, if it exists, consists of an object*

**D***and a natural isomorphism*

**colim**_{I}D ∈ C*of functors*

**Hom**_{C}(colim_{I}D, −) ≅ Hom_{CI}(D, (−)^{I})*. If all colimits of shape*

**C → Set***exist in*

**I***then this extends uniquely to an adjunction*

**C***. This explains the title of the previous lemma.]*

**colim**_{I}∶ C^{I}⇄ C ∶ (−)^{I}**Theorem (RAPL):**

Let * L ∶ C ⇄ D ∶ R* be an adjunction and consider a diagram

*of shape*

**D ∶ I → D***. If the diagram*

**I in D***has a limit cone*

**D ∶ I → D***then the composite diagram*

**Λ ∶ l**^{I}⇒ D*also has a limit cone, which is given by*

**R**^{I}(D) ∶ I → C*.*

**R**^{I}(Λ) ∶ R(l)^{I}⇒ R^{I}(D)**Proof:**

In this proof we will write * lim_{I}D ∶= l ∈ D*, and we will just assume that the limit object limI

*exists. Now we want to show that the following objects are isomorphic in*

**R**^{I}(D) ∈ C*. (We will ignore the data of the limit cone.)*

**C : R(lim**_{I}D) ≅ lim_{I}R^{I}(D)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 ∈ C**^{op}**Hom _{c}(c, R(lim_{I}D)) →^{~} Hom_{c}(L(c), lim_{I}D) (L ⊣ R)**

**→ ^{~} Hom_{D}I(L(c),D) (Diagonal ⊣ Limit)**

**→ ^{~} Hom_{D}I (L^{I} (c^{I}),D)**

**→ ^{~} Hom_{CI}(c,R^{I}(D))**

**→ ^{~} Hom_{c}(c,lim_{I}R^{I}(D)) (Diagonal ⊣ Limit)**

By composing these we obtain a family of bijections

**Hom _{c}(c,R(lim_{I}D)) →^{~} Hom_{c}(c,lim_{I} R^{I}(D))**

that is natural in * c ∈ C^{op}*. In other words, we obtain an isomorphism of hom functors

*in the category*

**H**_{R(limI(D))}≅ H_{limI}R^{I}(D)*. Then since the Yoneda embedding*

**Set**^{Cop}*is essentially injective (from the Embedding Lemma), we obtain an isomorphism of objects*

**H**_{(−)}: C → Set^{Cop}*in the category*

**R(lim**_{I}D) ≅ lim_{I}R^{I}(D)*.*

**C**