Dear All,

I hope all is well.

As you all know, following Samuel's Universal Mappings, Mac Lane interpreted product (A x B of factors A and B) as a universal mapping property (pp. 489-490), i.e., 1-1 correspondence between (pairs of) maps to the factors T --> A x T --> B and maps to the (corresponding) products T --> A x B, which is what we use to calculate products (pp. 6-7) in a category, with T as basic shape(s; yes, there is more, but skipped to get to the main point).  Isn't this 1-1 correspondence actually a part of the isomorphism: A^T x B^T ~ (A x B)^T?  To begin with, map objects, unlike those in the case of sets, involve nontrivial calculations (e.g.,. A^A, where A is the basic shape of arrow (* --> *) in the category of graphs), leave alone establishing isomorphisms.  Be that as it may, does this mean algebraic concepts defined as universal mapping properties are as much unique as they are isomorphic?  Even if it all sound word salad, there is still the question of how the 'for every' and 'there exists' in universal mapping definitions relate to isomorphisms.  In this context, one is reminded of 'there exists' left adjoint of 'substitution / composition' left adjoint of 'for every' (p. 246), except for the uniqueness of existence as in 'there exists a unique map' in universal mapping definitions.  Once we acknowledge isomorphisms in the universal mapping property definitions of category theory, then aren't categorical constructs within the confines of the category of groupoids?

I look forward to your corrections!

Thanking you,

Yours respectfully,

posina