Comma Category

Let , and be Categories Let and The comma category, written as is the category defined as follows

1. Objects are triples with , and
2. maps are pairs such that we have a Commutative Diagram

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
P(A) \arrow[r,"P(f)"] \arrow[d,"h"]
 & P(A') \arrow[d,"h'"] \\
Q(B) \arrow[r,"Q(g)"]
 & Q(B')
\end{tikzcd}
\end{document}

Special case

Let be a Functor. Let also be the functor from the trivial category to , sending the only element of to the object (by abuse of notation) We write for their comma category. This category has

1. Objects as pairs for and
2. Morphisms as morphisms such that we have a Commutative Diagram

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r, "h"] \arrow[dr,swap, "h'"] 
& GB \arrow[d, "Gf"] \\
& GB'
\end{tikzcd}
\end{document}

Theorem

Let be a Functor. Then specifying a left Adjoint for is equivalent to specifying an Initial object of the Comma Category for each .

Proof

Suppose has a left Adjoint . Let be the morphism corresponding to Then is Initial in . Let be an object in where A map in is in such that we have a Commutative Diagram

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r, "\eta_{A}"] \arrow[dr,swap, "f"] 
& GFA \arrow[d, "Gq"] \\
& GB
\end{tikzcd}
\end{document}

i.e.

But these correspond uniquely to

Thus exists and is unique.

Suppose for any we are given an initial object of This defines a Functor on objects . For any define to be the unique in Then is a morphism and functoriality of follows from uniqueness. The Adjunction sends each to the unique in . On the other hand, each is sent to and we can verify naturality.