A <representation of a functor> $(A,x)$ for $G$ says precisely that for every object $(B,y)$ of $(1\downarrow G)$ there is a unique arrow $f:A\to B$ with $Gf(x)=y$. This is exactly the <initial object> property for $(A,x)$ in that comma <category>. Conversely, an <initial object> supplies these unique arrows, hence the representing <bijections> $\mathcal C(A,B)\cong GB$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 18 / 5 / b / Solution

 Solution (source code)