This is Awodey exercise 7.12. I have difficulty understanding why the middle arrow is indeed unique.
Solution says that it is unique by the uniqueness of $h$, but I am stuck on how to deduce the uniqueness of the middle map is deduced from the uniqueness of $h$. Some more details or useful relative fact or lemmas, please? Thanks!
The solution:
If $\sf C\cong D$, then there are functors $F :{\sf C \rightleftarrows D} : G$ and natural isomorphisms $α :1_{\sf D} → FG$ and $β : GF →1_{\sf C}$. Suppose $\sf C$ has products, and let $D,D' ∈\sf D$ be given. We claim that $F(GD × GD')$ is a product object of $D$ and $D'$, with projections $α^{−1}_{\sf D} ◦ Fπ_1^{GD×GD'}$ and $α^{−1}_{\sf D} ◦ Fπ_2^{GD×GD'}$ . For suppose we have an object $Z$ and arrows $a : Z → D$ and $a' : Z → D'$ in $\sf D$. There is a unique $h : GZ → GD ×GD' ∈\sf C$ such that $π^{GD×GD'}_1 ◦h = Ga$ and $π^{GD×GD'}_2 ◦ h = Ga'$. Then the mediating map in $\sf D$ is $Fh◦ α_Z$. We can calculate $α^{−1}_D ◦Fπ^{GD×GD'}_1 ◦Fh◦α_Z = α^{−1}_D ◦F(π^{GD×GD'}_1 h)◦α_Z = α^{−1}_D ◦FGa◦α_Z = α^{−1}_D ◦α_D ◦a = a$ and similarly for the second projection. Uniqueness of the map $Fh◦α_Z$ follows from that of $h$.