MATH.5:5.3 - Interpret paths with different kinds of values

Take source objects A and B, with generators u:A -> B and v:B -> A. Interpret A as the integers and B as pairs of integers. Assign u_target(n)=(n,0) and v_target(n,m)=n. The empty path at A becomes the identity on integers; the empty path at B becomes the identity on integer pairs.

Extension gives E(u;v)(n)=n, whereas E(v;u)(n,m)=(n,0). The first composite therefore equals E(id_A). The second differs from E(id_B): it sends (7,5) to (7,0). The interpretation can descend through the source equation u;v=id_A, together with the equations generated from it by permitted composition. It keeps the information that the return path loses the second component.

Now require v;u=id_B as well. The same assignment fails that new equation. Keep the free-path interpretation or the first quotient, change the assigned maps, or change the required identification. Both composites are valid paths; the failed assertion concerns their values, not whether they can be formed. This distinction lets a representation carry construction followed by recovery without claiming recovery in both directions.