Why are pushouts the right tool in these setups

Question

$\newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\cod}{\operatorname{cod}} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\colim}{colim}$The question is about two pushout constructions G.M. Kelly is using in his paper A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on (§14.1) to construct colimts and adjoints in a comma category setting. Sadly, he neither gives a motivation for these constructions, nor does he prove they do what they are supposed to do. And I can't figure out why pushouts are the right tool in this context.

I don't know weather there is a common answer for both constructions, but they appear side by side in a similar setting, so I figured I might as well put them into the same question.

The first one is

Given a category $\cat{C}$ and an endofunctor $T\colon\cat{C}\to\cat{C}$, a diagram $D\colon I\to T\downarrow\cat{C}$ is given by two functors $X,Y\colon I\to\cat{C}$, and a natural transformation $\kappa\colon T\circ X\Rightarrow Y$. Then the colimit $\colim_I D$ is given by $(\colim X,f,x)$, where $f$ and $x$ are given by the pushout $$ \require{AMScd} \begin{CD} \colim T\circ X@>{\colim\kappa}>> \colim Y\\ @V{\tilde T}VV @VhVV \\ T\colim X@>{f}>> x \end{CD} $$ where $\tilde T$ is the canonical comparison map.

whereas the second is

Given a category $\cat{C}$, two endofunctors $T,T'\colon\cat{C}\to\cat{C}$ and a natural transformation $\alpha\colon T'\to T$, we obtain a functor $\alpha^*\colon T\downarrow\cat{C}\to T'\downarrow\cat{C}$, which sends $(x,f,y)$ to $(x,f\circ\alpha_x,y)$. This functor has a left adjoint $\alpha_*\colon T'\downarrow \cat{C}\to T\downarrow \cat{C}$ which sends $(x',f',y')$ to $(x',\bar f' ,\bar y' )$, where $\bar f'$ and $\bar y'$ are given by the pushout $$ \require{AMScd} \begin{CD} T'(x)@>{f'}>> y\\ @V{\alpha_x}VV @V\hat f' VV \\ T(x)@>{\bar f'}>> \bar y'. \end{CD} $$

I'm not asking for a calculation why these constructions do what they are supposed to do, but rather some kind of motivation/explanation, why the author considers using pushouts in the first place and why they are supposed to work in this setting.

[Edit] Ok, so the second diagram seems to use what the nlab calls a cobase change along every component of $\alpha$. The respective nlab article itself isn't very useful, though.

Answer 1

$\DeclareMathOperator\colim{colim}\newcommand\uHom{\underline{\operatorname{Hom}}}$For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times \mathrm{id}}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \uHom(I, A\times_C B) \to A \times_C B $$ where $\uHom$ is the internal hom in $\mathrm{Cat}$. By the assumption one sees that this is identified with the functor on $$ \uHom(I, A) \times_{\uHom(I, C)} \uHom(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.