I’d appreciate very much if you could address the following question:
Given two composable functions [domain (one) = codomain (other)]: the unique function ‘i’ with empty set E as domain and a set X as codomain, and a function ‘f’ with the set X as domain and a set Y as codomain [domain (f) = codomain (i)], how is the composite function ‘fi’ of the two functions i and f defined?
The above question came about during the course of studying maps as a means of indicating various figures in the objects of a category. For example, we can use maps (functions) with terminal object (singleton set) as domain to indicate elements of codomain objects in the category of sets. Along similar lines, we can use graph maps with a domain graph of one arrow and two dots (source dot, target dot) to indicate arrows of codomain graphs in the category of (irreflexive directed multi-) graphs (Conceptual Mathematics textbook of Lawvere & Schanuel, page 215).
Objects of the category of graphs have, in addition to arrows, dots. It is in trying to use maps in the category of graphs (Conceptual Mathematics, pp. 141-2) with domain graph with exactly one dot and zero arrows to indicate dots in codomain graphs that I encountered the problem of the composite map (in the category of sets) fi, with empty set as domain of the function i and with domain (f) = codomain (i).
I eagerly look forward to any corrections and clarifications that you, your time permitting, could provide.
Thank you, vrayudu