$\newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\cod}{\operatorname{cod}} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\colim}{colim}$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.