Help me understand the diagram category

Question

In category theory a diagram in a category $\textbf{C}$ is a functor $D:\textbf{J}\to\textbf{C}$, where $\textbf{J}$ is a small category. I want to understand categories of diagrams. The only definition I can find online after an initial search is the following:

Definition. The category $\text{Diag}(\textbf{C})$ has as objects diagrams in $\textbf{C}$ and as morphisms from $D:\textbf{J}\to\textbf{C}$ to $D^\prime:\textbf{J}^\prime\to\textbf{C}$ a functor $R:\textbf{J}\to\textbf{J}^\prime$ together with a natural transformation $\rho:\textbf{J}\Rightarrow\textbf{J}^\prime\circ R$.

My conceptual issue involves the natural transformation $\rho$. Can you explain what it means to compose a category with a functor as in $\textbf{J}^\prime\circ R$? If there is a more clear definition can you give it?

Answer 1

There are two particularly useful versions of this category. $\mathrm{Diag}_l(C)$ has as objects the diagrams $D:J\to C$ and as morphisms $(D:J\to C)\to(D’:J’\to C’)$ the pairs $(R,\rho)$ of a functor $R:J’\to J$ and a natural transformation $D\circ R\to D’.$ The alternative $\mathrm{Diag}_c(C)$ is just the same except that the functor part $R$ of a morphism goes $J\to J’$ and thus $\rho$ must be modified to go $D\to D’\circ R.$

The reason these two categories are particularly interesting is that, if $C$ is complete, then the operation of taking limits is a functor from $\mathrm{Diag}_l(C)$ to $C$; and similarly for $\mathrm{Diag}_c(C)$ and colimits.

Just because this is my favorite reference in category theory…these diagram categories are defined (correctly) in Paragraph 23 of Eilenberg and Mac Lane’s 1942 paper “The general theory of natural equivalences,” where the functoriality of the limit and the colimit is also proven. That’s the original paper inventing category theory!

Answer 2

In nLab they define the category of $\bf J$-shaped diagrams in $\bf C$ simply as functor category $\mathrm{Funct}(\bf J,\bf C)$ where objects are functors and morphisms natural transformations (see here).

Here you have two shapes $\bf J$ and $\bf J'$, so you can't look at natural transformation between functors $D\colon \bf J\to \bf C$ and $D'\colon \bf J'\to \bf C$ because domains of $D$ and $D'$ are not the same but, if you have a functor $R\colon \bf J\to \bf J'$, then you can look at natural transformations between functors $D$ and $D'\circ R$.

I suspect the definition should be that a morphism in category $\mathrm{Diag}(\bf C)$ between functors $D\colon \bf J\to \bf C$ and $D'\colon \bf J'\to \bf C$ is a pair $(R,\rho)$ where $R\colon\bf J\to \bf J'$ is a functor and $\rho\colon D\Rightarrow D'\circ R$ is natural transformation.