To prove the RAPL theorem we must first translate the definition of limit/colimit into a language that is compatible with the definition of adjoint functors.

Recall that a diagram is a functor * D ∶ I → C* from a small category

*. If*

**I***is locally small then we have a locally small category*

**C***consisting of diagrams and natural transformations between them. For each object*

**C**^{I}*we also have the constant diagram*

**c ∈ C***that sends each object*

**c**^{I}∶ I → C*to*

**i ∈ Obj(I)***and each arrow*

**c**^{I}(i) ∶= c*to*

**δ ∈ Arr(I)***.*

**c**^{I}(δ) ∶= id_{c}It is a general phenomenon that many categorical properties of * C^{I}* are inherited from

*. The next lemma collects a few of these properties that we will need later.*

**C**Diagram Lemma. Fix a small category * I* and locally small categories

*. Then:*

**C, D**(i) For any category * C*, the mapping

*defines a fully faithful functor*

**c ↦ c**^{I}*which we call the diagonal embedding.*

**(−)**^{I}∶ C → C^{I}(ii) For any functor * F ∶ C → D* the mapping

*defines a functor*

**F**^{I}(D)∶= F ○ D*with the property that*

**F**^{I}∶ C^{I}→ D^{I}*.*

**F (−)**^{I}= F^{I}((−)^{I})(iii) Any adjunction * L ∶ C ⇄ D ∶ R* induces an adjunction

*That is, we have a natural isomorphism of bifunctors*

**L**^{I}∶ C^{I}⇄ D^{I}∶ R^{I}**Hom _{CI} (−, R^{I}(−)) ≅ Hom_{DI} (L^{I}(−), −)**

from * (C^{I})^{op} × D^{I} to Set*.

(iv) In particular, naturality in * D^{I}* tells us that for all objects

*and all natural transformations*

**l ∈ C***we have a commutative square:*

**Λ ∶ l**^{I}⇒ D**Proof:**

(i): For any arrow * α ∶ c_{1} → c_{2} in C* we want to define a natural transformation of diagrams

*, and there is only one way to do this. Since*

**α**^{I}∶ c^{I}_{1}⇒ c^{I}_{2}*and*

**(c**^{I}_{1})_{i}= c_{1}*, the arrow*

**(c**^{I}_{2})_{i}= c_{2}**∀ i ∈ I***must be defined by*

**(α**^{I})_{i}∶= (c^{I}_{1})_{i}→ (c^{I}_{2})_{i}*. Then for any arrow*

**(α**^{I})_{i}∶= α*we have*

**δ ∶ i → j in I***and*

**c**^{I}_{1}(δ) = id_{c1}*, so that*

**c**^{I}_{2}(δ) = id_{c2}**(α ^{I})_{i} ○ (c^{I}_{1}) (δ) = (α ○ id_{c1}) = (id_{c2} ○ α) = (c^{I}_{2})_{i} (δ) ○ (α^{I})_{i}**

and hence we obtain a natural transformation **α ^{I}**

*. The assignment*

**∶ c**^{I}_{1}⇒ c^{I}_{2}*is functorial since for all arrows*

**α ↦ α**^{I}*such that*

**α, β***exists and*

**α ○ β***we have*

**∀ i ∈ I***,*

**(α ○ β)**^{I}_{i}= α ○ β = (α^{I})_{i}○ (β^{I})_{i}= (α^{I}○ β^{I})_{i}and hence * (α ○ β)^{I} = α^{I} ○ β^{I}*. Finally, note that we have a bijection of hom sets

*given by*

**Hom**_{C}(c_{1}, c_{2}) ↔ Hom_{CI}(c^{I1}, c^{I2})*, and hence the functor*

**α ↔ α**^{I}*is fully faithful.*

**(−)**^{I}∶ C → C^{I}(ii): Let * F ∶ C → D* be any functor. Then for any diagram

*we obtain a diagram*

**D ∶ I → C***by composition:*

**F**^{I}(D) ∶ I → D*. This assignment is functorial in*

**F**^{I}(D) ∶= F ○ D*. To see this, consider any natural transformation*

**D ∈ C**^{I}*in the category*

**Φ ∶ D**_{1}⇒ D_{2}*. Then for any arrow*

**C**^{I}*we can apply*

**δ ∶ i → j in I***to the naturality square for*

**F***to obtain another commutative square:*

**Φ**If we define * F^{I}(Φ)_{i} ∶= F(Φ_{i}) ∀ i ∈ I* then this second commutative square says that

*is a natural transformation in*

**F**^{I}(Φ) ∶ F^{I}(D_{1}) ⇒ F^{I}(D_{2})*. If*

**D**^{I}*and*

**Φ***are two arrows (natural transformations) in*

**Ψ***such that*

**C**^{I}*is defined, then*

**Φ ○ Ψ***we have*

**∀ i ∈ I***and hence*

**F**^{I}(Φ ○ Ψ)_{i}= F((Φ ○ Ψ)_{i}) = F(Φ_{i}○ Ψ_{i}) = F(Φ_{i}) ○ F(Ψ_{i}) = F^{I}(Φ)_{i}○ F^{I}(Ψ)_{i}= (F^{I}(Φ) ○ F^{I}(Ψ))_{i}*. Thus we have defined a functor*

**F**^{I}(Φ ○ Ψ) = F^{I}(Φ) ○ F^{I}(Ψ)*. Finally, note that*

**F**^{I}∶ C^{I}→ D^{I}*, and*

**∀ i ∈ I, c ∈ C***we have*

**α ∈ Arr(C)****F ^{I}(c^{I})_{i} = F((c^{I})_{i}) = F(c) = ((F(c))^{I})_{i} F^{I}(α^{I})_{i} = F((α^{I})_{i}) = F(α) = ((F(α))^{I})_{i}**

and hence we have an equality of functors * F^{I}((−)^{I}) = F (−)^{I}* from

**C to D**^{I}(iii): Let * L ∶ C ⇄ D ∶ R* be any adjunction. We will denote each bijection

*by*

**Hom**_{C}(−,R(−)) ↔ Hom_{C}(L(−), −)*, so that*

**φ ↦ φ**^{–}*. Now we want to define a natural family of bijections*

**φ**^{=}= φ

**Hom**_{CI}(−, RI (−)) ≅ Hom_{DI}(L^{I}(−), −)To do this, consider diagrams * C ∈ C^{I}, D ∈ D^{I}*, and a natural transformation

*. Then for each index*

**Φ ∶ C ⇒ R**^{I}(D)*we have an arrow*

**i ∈ I***, which determines an arrow*

**Φ**_{i}∶ C(i) → R(D(i))*by adjunction. The arrows*

**Φ**^{–}_{i}∶ L(C(i)) → D(i)*assemble into a natural transformation*

**Φ**^{–}_{i}*. To see this, consider any arrow*

**Φ**^{–}∶ L^{I}(C) ⇒ D*. Then from the naturality of*

**δ ∶ i ∈ j in I***and the adjunction*

**Φ***we have*

**L ⊣ R*** D(δ) ○ Φ^{–}_{i} = (R(D(δ)) ○ Φ_{i})^{–} * naturality of

**L ⊣ R*** = (Φ_{j} ○ C(δ))^{–}* naturality of

**Φ**= * Φ_{j}^{–} ○ L(C(δ))* naturality of

**L ⊣ R**as desired. In a similar way one can check that for each natural transformation * Ψ ∶ L^{I}(C) ⇒ D*, the arrows

*assemble into a natural transformation*

**Ψ**_{i}∶ C(i) → R(D(i))*. Thus we have established the desired bijection of hom sets*

**Ψ ∶ C ⇒ R**^{I}(D)*.*

**Hom**_{CI}(C, R^{I}(D)) ↔ Hom_{DI}(L^{I}(C), D)To prove that this bijection is natural in * (C, D) ∈ (C^{I})^{op} × D^{I}*, consider any pair of natural transformations

*in*

**Γ∶ C**_{2}⇒ C_{1}*and*

**C**^{I}*. We need to show that a certain cube of functions commutes. For a fixed diagram C ∈ C*

**∆ ∶ D**_{1}⇒ D_{2}in D^{I}^{I}the following square commutes:

First, recall that the natural transformation * R^{I}(∆) ∶ R^{I}(D_{1}) ⇒ R^{I}(D_{2})* is defined pointwise by

*. Now consider any*

**R**^{I}(∆)_{i}∶= R(∆_{i}) ∶ R(D_{1}(i)) → R(D_{2}(i))*. The naturality of the original adjunction tells us that*

**Φ ∶ C ⇒ R**^{I}(D_{1})*, and hence we have*

**(R(∆**_{i}) ○ Φ_{i})^{–}= ∆i ○ Φ_{i}^{–}**((R ^{I}(∆) ○ Φ)_{i})^{–} = (R^{I}(∆)_{i} ○ Φ_{i})^{–}**

**= (R(∆ _{i}) ○ Φ_{i})^{–}**

**= ∆ _{i} ○ Φ_{i}^{–}**

**= (∆ ○ Φ ^{–})_{i}**

* ∀ i ∈ I*. By definition this means that

*, and hence the desired square commutes. It remains only to check that the cube is natural in*

**(R**^{I}(∆) ○ Φ)^{–}= ∆ ○ Φ^{–}*. This follows from a similar pointwise computation.*

**(C**^{I})^{op}(iv): Now fix an element * l ∈ C*, a diagram

*, and a natural transformation*

**D ∈ D**^{I}*. By substituting*

**Λ ∶ l**^{I}⇒ D*, and*

**C = R(l)**^{I}, D_{1}= l^{I}, D_{2}= D*into the above commutative square and using part (ii), we obtain the commutative square from the statement of the lemma. In particular, following the identity arrow*

**∆ = Λ***around the square in two ways gives*

**id**^{I}_{R(l)}**(R ^{I}(Λ) ○ id^{I}_{R(l)})^{–} = Λ ○ (id^{I}_{R(l)})^{–}**

Finally, one can check pointwise that * (id^{I}_{R(l)})^{–} = ((id_{R(l)})^{I})^{–}* and hence we obtain the identity

**(R ^{I}(Λ) ○ id^{I}_{R(l)})^{–} = ((id_{R(l)})^{I})^{–}**

Now we will reformulate the definition of limit/colimit in terms of adjoint functors. If all limits/colimits of shape* I* exist in some category

*then it turns out (surprisingly) that we can think of limits/colimits as right/left adjoints to the diagonal embedding*

**C**

**(−)**^{I}∶ C → C^{I}: colim_{I}⊣(−)^{I}⊣ lim_{I}In the next section/part’s lemma we will prove something slightly more general. We will characterize a specific limit/colimit of shape I, without assuming that all limits/colimits of shape * I* exist.