MHB Help w/ Awodey Exercise 5, Ch. 1: Functor from Slice Category to Any Category

[Math Amateur]

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Chapter 1: Categories

I need some help in order to make a meaningful start on Awodey Exercise 5, Chapter 1 Awodey Exercise 5, Chapter 1 reads as follows:
View attachment 8392
I am unable to make a meaningful start on this exercise ... can someone please help me to formulate a solution to the above exercise ...I would like to utilise the exact form of Awodey's definition of functor (see below) in the solution to the exercise ...
Help will be much appreciated ...

Peter=======================================================================================
It may well help readers of the above post to have access to Awodey's definition of a slice category ... so I am providing the same ... as follows:View attachment 8393
View attachment 8394

It may also help readers of the above post to have access to Awodey's definition of a functor ... so I am providing the same ... as follows:
View attachment 8395
View attachment 8396
Hope that helps ...

Peter

 

[steenis]

Given a category $\mathscr{C}$, fix an object $C \in \mathscr{C}$

Define the catergory $\mathscr{C}/C$ as follows:
Objects: the objects of $\mathscr{C}/C$ are the arrows $f:A \rightarrow C$ in $\mathscr{C}$
Arrows: the arrow $a:f \rightarrow g$ in $\mathscr{C}/C$ between two objects $f:A \rightarrow C$ and $g:B \rightarrow C$ in $\mathscr{C}/C$ is an arrow $a:A \rightarrow B$ in $\mathscr{C}$, such that $g \circ a = f$

Define the category $\mathscr{C}^{\rightarrow}$ as follows:
Objects: the objects of $\mathscr{C}^{\rightarrow}$ are the arrows of $\mathscr{C}$
Arrows: the arrow $h:f \rightarrow g$ in $\mathscr{C}^{\rightarrow}$ between two objects $f:A \rightarrow X$ and $g:B \rightarrow Y$ in $\mathscr{C}^{\rightarrow}$ is a pair $h=(h_1,h_2)$, where $h_1:A \rightarrow B$ and $h_2:X \rightarrow Y$ are arrows in $\mathscr{C}$, such that $h_2 \circ f = g \circ h_1$

Define the forgetful functor $U: \mathscr{C}/C \rightarrow \mathscr{C}$ as follows:
Objects: if $f:A \rightarrow C$ is an object in $\mathscr{C}/C$, define $Uf=A$, which is an object in $\mathscr{C}$
Arrows: if $a:f \rightarrow g$ is an arrow in $\mathscr{C}/C$ between the objects $f:A \rightarrow C$ and $g:B \rightarrow C$ in $\mathscr{C}/C$, define $Ua=a: A \rightarrow B$, which is an arrow in $\mathscr{C}$
Notice that $Ua:Uf \rightarrow Ug$ and show that $U$ a functor $U: \mathscr{C}/C \rightarrow \mathscr{C}$

Define the functor $F: \mathscr{C}/C \rightarrow \mathscr{C}^{\rightarrow}$ as follows:
Objects: if $f:A \rightarrow C$ is an object in $\mathscr{C}/C$, then $Ff=f:A \rightarrow C$ which is an object in $\mathscr{C}^{\rightarrow}$
Arrow: if $a:f \rightarrow g$ is an arrow in $\mathscr{C}/C$ between the objects $f:A \rightarrow C$ and $g:B \rightarrow C$ in $\mathscr{C}/C$, i.e., $a:A \rightarrow B$, then $Fa:Ff \rightarrow Fg$ is the arrow $Fa=(a, 1_C)$ in $\mathscr{C}^{\rightarrow}$ between the objects $Ff= f:A \rightarrow C$ and $Fg= g:B \rightarrow C$ in $\mathscr{C}^{\rightarrow}$. Notice $g \circ a = f \circ 1_C$.
Show that $F: \mathscr{C}/C \rightarrow \mathscr{C}^{\rightarrow}$ is a functor

Now $dom \circ F(f:A \rightarrow C) = dom(f:A \rightarrow C) = A$
And $U(f:A \rightarrow C) = A$

 

[steenis]

I am unable to make a meaningful start on this exercise ... can someone please help me to formulate a solution to the above exercise ...

Can you see how I approached this problem ?

I should have made diagrams in my solution, but that is very cumbersome in Latex, you should make them when studying this problem.