# Why are pushouts the right tool in these setups

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

 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.

$$\DeclareMathOperator\colim{colim}\newcommand\uHom{\underline{\operatorname{Hom}}}$$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.