I'm reading Goldblatt's "Topoi The Categorial Analysis of Logic". In it he introduces products early on, using a diagram like, $$\begin{array}{center} \; d \\ \; f \swarrow \downarrow m \searrow g \\ \; a \longleftarrow \; a \times b \longrightarrow b \\ \end{array}$$ $$\begin{array}{center} \; pr_a && pr_b \end{array}$$ where $$ m=<f,g> $$ is unique. (the arrow m is denoted by a broken arrow for uniqueness, and the diagram would probably look better if the diagonal arrows were longer and I knew how to do spaces, but I didn't know how to do all that this being my first Q here on these forums, much less my first diagram).
(Before continuing, I like to call the object at top (i.e. d in my diagram above), the "auxilary object". Which afaik could be ANY object in our category that has arrows to the "factor objects": And for two given objects in defining their product, I call them "factor objects" (in reference to the class of isomorphic product objects and their respective projections.) So a and b are factor objects in the diagram above.)
As he continues, it looks as though the unique arrow, called the "product arrow", is usually denoted as an ordered pair of the given arrows from the auxilary object to each of the factor objects, being called the product of the mentioned arrows. So for example, again $$ m=<f,g> $$
The author mentions that a product of two objects is not necessarily unique, and in fact goes on to prove that all products are isomorphic. In the proof he flips the diagram above yielding, $$\begin{array}{center} \; a \times b \\ \; pr_a \swarrow \downarrow n \searrow pr_b \\ \; a \longleftarrow \; d \longrightarrow b \\ \end{array}$$ where this time the auxilary object is the product $$ a \times b $$ and this time an arbitrary product object d, of a and b shows up instead in the middle bottom. But now the product arrow n, is $$ n=<pr_a,pr_b> $$ again staying with the notation that the product arrow is written as an ordered pair of the auxilary object's given arrows to the given factor objects.
In one of the following exercises the author uses the same notation in asking the reader to prove that $$ <pr_a,pr_b>=id_{a\times b} $$ for the product-defining diagram in which both the auxilary object and the product are the same, $$ a \times b $$ for: $$\begin{array}{center} \; a \times b \\ \; pr_a \swarrow \downarrow m \searrow pr_b \\ \; a \longleftarrow \; a \times b \longrightarrow b \\ \end{array}$$ $$\begin{array}{center} \; pr_a && pr_b \end{array}$$ Ok I understand the proof and was able to prove the exercise, but neither tells me:
OK NOW my question:
Is this merely an abuse of notation, using the same symbol for possibly distinct arrows in different diagrams?
OR
Despite the definition of product being "up to isomorphism", it actually happens to always be unique? Iow, products actually ARE unique, and this uniqueness is implied by the category axioms and the defintion of product (which initially is at least up to iso)? In which case I missed the other point of the exercise and didn't actually finish it (and still can't either way, considering I'm asking this now).