A functor continuous with respect to a cylinder In the erratum of their paper: “Categories of continuous functors, I”, (Journal of Pure and Appplied Algebra 2 (1972) 169–191), the authors, P.J. Freyd and G.M. Kelly, define the continuity of a functor $T: \mathcal{C} \to \mathcal{A}$ relative to a “cylinder” $\alpha$ as follows:

*

*First, they define a “cylinder” as the 4-uple made of $(J, P, Q, \alpha)$ where $J \colon \mathcal{K} \to \mathcal{L}$, $P \colon \mathcal{L} \to \mathcal{C}$ and $Q \colon \mathcal{K} \to \mathcal{C}$ are functors, and $\alpha \colon PJ \xrightarrow{\cdot} Q \colon \mathcal{K} \to \mathcal{C}$ is a natural transformation. (They refer to the cylinder $(J, P, Q, \alpha)$ writing it simply as $\alpha$).


*Then, they define: $T$ is continuous relative to the cylinder $\alpha$ if and only if the composite
$$
  \lim\,TP \to \lim\, TPJ \to \lim\, TQ
  \tag{1}
$$
is an isomorphism, “the first map being the canonical one and the second being $\lim\, T$”.
My question is:
What exactly are those morphisms appearing in equation (1)?
For example, for the functor $T \colon \mathcal{C} \to \mathcal{A}$ the standard definition for the limit of $T$ is a pair $(\overline {\lim\,T}, \overline{T})$ where $\overline{\lim\,T} \in \mathrm{Obj}_{\mathcal{A}}$, and $\overline{T} \colon (\overline{\lim\,T})_{\mathcal{C}} \xrightarrow{\cdot} T \colon \mathcal{C} \to \mathcal{A}$.
Then, in what sense is the second morphism in eq. (1), $\lim\, TPJ \to \lim\, TQ$, related to the limit of $T$, understood here as being the natural transformation $\overline{T}$?
And what is the “canonical map” $\lim\,TP \to \lim\, TPJ$?
 A: Whenever you have a diagram $X\colon\mathcal{I}\rightarrow\mathcal{C}$ and a morphism $T\colon\mathcal{J}\rightarrow\mathcal{I}$, there is a canonical morphism $\lim X\rightarrow\lim XT$. This, by the universal property, corresponds to giving compatible morphisms $\lim X\rightarrow XT(j)$ for each object $j\in\mathcal{J}$ and the canonical choice is taking the canonical projections. The first morphism is a special case of this.
Whenever you have diagrams $X,Y\colon\mathcal{I}\rightarrow\mathcal{C}$ and a natural transformation $\alpha\colon X\Rightarrow Y$, there is a canonical morphism $\lim\alpha\colon\lim X\rightarrow\lim Y$. This, by the universal property, corresponds to giving compatible morphisms $\lim X\rightarrow Y(i)$ for each object $i\in\mathcal{I}$ and the canonical choice is taking the composition $\lim X\rightarrow X(i)\rightarrow Y(i)$ of the canonical projection and $\alpha(i)$. The second morphism is a special case of this.
A: There are two kinds of construction going on here (that are probably special cases of one general construction).

*

*Let $P \colon \mathcal{C} \to \mathcal{D}$ be a functor and let $J \colon \mathcal{B} \to \mathcal{C}$ be another functor.
There exists a unique morphism
$$
  r \colon \lim P \to \lim PJ
$$
with $\mathrm{pr}_B ∘ r = \mathrm{pr}_{J(B)}$ for every object $B$ of $\mathcal{B}$, i.e., such that the diagram
$$
  \require{AMScd}
  \begin{CD}
    \lim P                     @> r >> \lim PJ \\
    @V \mathrm{pr}_{J(B)} VV           @VV \mathrm{pr}_B V \\
    PJ(B)                      @=      PJ(B)
  \end{CD}
$$
commutes for every object $B$ of $\mathcal{B}$.
For some intuition on this, one can consider the case that the category $\mathcal{C}$ is discrete, $\mathcal{B}$ is a subcategory of $\mathcal{C}$, and $J$ is the inclusion functor from $\mathcal{B}$ to $\mathcal{C}$.
Then $\lim P = \prod_C P(C)$ and $\lim PJ = \prod_B PJ(B) = \prod_B P(B)$.
The morphism $r$ goes from the larger product $\prod_C P(C)$ into the smaller product $\prod_B P(B)$ by throwing some of the factors away.
Suppose more generally that $\mathcal{B}$ is a subcategory of $\mathcal{C}$ and that $J$ is the inclusion functor from $\mathcal{B}$ to $\mathcal{C}$.
The functor $P$ describes a diagram in $\mathcal{D}$.
The term $\lim P$ is the limit of the entire diagram, whereas $\lim PJ$ is the limit over only a part of the diagram.


*Let $P, Q \colon \mathcal{C} \to \mathcal{D}$ be two functors and let $α \colon P \Rightarrow Q$ be a natural transformation.
There exists a unique morphism
$$
  \lim α \colon \lim P \to \lim Q
$$
with $\mathrm{pr}_C ∘ \lim α = α_C ∘ \mathrm{pr}_C$ for every object $C$ of $\mathcal{C}$, i.e., such that the diagram
$$
  \require{AMScd}
  \begin{CD}
    \lim P                @> \lim α >>   \lim Q              \\
    @V \mathrm{pr}_C VV                  @VV \mathrm{pr}_C V \\
    P(C)                  @> α_C >>      Q(C)
  \end{CD}
$$
commutes for every object $C$ of $\mathcal{C}$.
This construction is essentially how the functoriality of
$$
  \lim \colon \mathrm{Fun}(\mathcal{C}, \mathcal{D}) \to \mathcal{D}
$$
works.
As a specific example, suppose that $\mathcal{C}$ is discrete with two objects, named $1$ and $2$.
Then $\lim P = P(1) × P(2)$ and $\lim Q = Q(1) × Q(2)$, and $\lim α = α_1 × α_2$.
In your specific case, the morphism from $\lim TP$ to $\lim TPJ$ comes from the first construction.
The morphism from $\lim TPJ$ to $\lim TQ$ comes from the second construction and the whiskered natural transformation $T α \colon TPJ \Rightarrow TQ$.
