From Categories for the Working Mathematician:
Question: Wouldn't universality imply the existence of TWO $h$ -- say $(h_1,h_2)$ -- s.t. $h_1 \circ i = f$ and $h_2 \circ j = g$?
That is, if $u = (i,j): (a,b) \rightarrow (c,c)$, then shouldn't universality imply that for any $n = (f,g): (a,b) \rightarrow (d,d)$, we have that there exists a unique $(h_1, h_2) : (c,c) \rightarrow (d,d)$ with $h_1 \circ i = f$ and $h_2 \circ j = g$? What reason is there to think that $h_1 = h_2$?
Or, if I am right, is Mac Lane just saying that the universality of the codproduct is distinct from the usual notion of universality used elsewhere in category theory?