ra.rings and algebras - Universal functors according to Cohn.

I have the 1965 edition of Cohn's book, which apparently differs from the one you cite, since what you refer to as Proposition 1.1 is Theorem 1.2 in my copy. Nevertheless, I hope the following comments may be useful.

First, about the mysterious "morphisms" between objects that live in different categories, Cohn writes (near the top of page 110 in my copy) "we generally refer to the elements of $F(A,a)$ as the admissible morphisms of the representation. I believe this explains the anomaly that bothered you.

Let me also mention that this overloading of the word "morphism" is not as silly as it might at first appear. There is a natural way to "glue together" two categories $mathcal L$ and $mathcal K$ using a functor $F$ as in your question. Start with the disjoint union of $mathcal L$ and $mathcal K$, and then add Cohn's admissible morphisms as actual morphisms from an object of $mathcal L$ to one of $mathcal K$. Of course, having added these morphisms, you need to define compositions involving them. But $F$ tells you how to do that: If $rho:Ato a$ is one of the new morphisms, you want to pre-compose it with morphisms $lambda:Bto A$ of $mathcal L$ and to post-compose it with morphisms $kappa:ato b$ of $mathcal K$. For the former, act on $rho$ by $F(lambda,1_a)$; for the latter, use $F(1_A,kappa)$. The functoriality of $F$ ensures that this definition of composition for the new morphisms plus the pre-existing composition laws of $mathcal L$ and $mathcal K$ satisfy the associative law.

I conjecture that this construction is the "join" mentioned in Herman Stel's answer, but I haven't checked "Higher Topos Theory" to make sure. I do, however, assure you that this construction was the mental picture of $F$ that came naturally to my mind when I started reading your question.