I am reading "Topoi - The Categorial Analysis Of Logic" by Robert Goldblatt, but I'm struggling with a detail - to quote:
"Notice that we said a product of $a$ and $b$, not the product . This is because $a \times b$ is only defined up to isomorphism. For suppose $(p \colon d \rightarrow a, q \colon d \rightarrow b)$ also satisfies the definition of "a product of $a \times b$" and consider the diagram:
..."
Now, I get most of it, but how do I see that $\left\langle pr_a,pr_b \right\rangle$ goes from $a \times b$ to $d$?
There is a overloading of the operator $\langle {-},{-} \rangle$ here.
The first one in the instance $\langle p,q \rangle$ comes from the natural isomorphism $$ \hom(-,a) \times \hom(-,b) \underset\simeq{\overset{\langle-,-\rangle} \to} \hom(-,a\times b) $$ given by the fact that $a\times b$ is a product of $a$ and $b$.
The second one in the instance $\langle \mathrm{pr}_a,\mathrm{pr}_b \rangle$ comes from the natural isomorphism $$ \hom(-,a) \times \hom(-,b) \underset\simeq{\overset{\langle-,-\rangle} \to} \hom(-,d) $$ given by the fact that $d$ is a product of $a$ and $b$.
To be perfectly rigorous, the authors should have subscripted each instance: $\langle p, q \rangle_{(a\times b,\mathrm{pr}_a,,\mathrm{pr}_b)}$ and $\langle \mathrm{pr}_a,\mathrm{pr}_b \rangle_{(d, p, q)}$. But it is heavy and usually the subscript is inferable from context. The same thing appears in many area of mathematics: for example we use the same (absence of) symbol for the multiplication of many many groups.