RCF1: Theories of PR Maps and Partial PR Maps
  We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd's uniqueness of the initialised iterated. Free Variables are introduced - formally - as another names for projections. Predicates \chi: A -> 2 admit interpretation as (formal) Objects {A|\chi} of a surrounding Theory PRA = PR + (abstr) : schema (abstr) formalises this predicate abstraction into additional Objects. Categorical Theory P\hat{R}_A \sqsupset PR_A \sqsupset PR then is the Theory of formally partial PR-maps, having Theory PR_A embedded. This Theory P\hat{R}_A bears the structure of a (still) diagonal monoidal category. It is equivalent to "the" categorical theory of \mu-recursion (and of while loops), viewed as partial PR maps. So the present approach to partial maps sheds new light on Church's Thesis, "embedded" into a Free-Variables, formally variable-free (categorical) framework. 
Michael Pfender