\documentclass{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
  picture U, XY, X, Y, Z;
  z1 = -z2 = (-61, 42);
  U  = thelabel("$U$", z1);
  XY = thelabel("$X\times_ZY$", origin);
  X  = thelabel("$X$", (x2, 0));
  Y  = thelabel("$Y$", (0, y2));
  Z =  thelabel("$Z$", z2);

  forsuffixes @=U, XY, X, Y, Z: draw @; endfor

  ahangle := 20;
  vardef connect@#(expr s, a, b) = 
    curved_connect@#(s, a, b, center b - center a)
  enddef;
  vardef curved_connect@#(expr s, a, b, d) = 
    save line, mark; path line; picture mark;
    line = center a {d} .. center b;
    interim bboxmargin := 4;
    drawarrow line cutbefore bbox a cutafter bbox b;
    mark = thelabel@#("$\scriptstyle " & s & "$", 
                                       point 1/2 of line);
    interim bboxmargin := 1;
    unfill bbox mark; draw mark;
  enddef;

  connect.bot("p", XY, X);
  connect.rt ("q", XY, Y);
  connect.top("g", Y, Z);
  connect.lft("f", X, Z);
  curved_connect.urt("x", U, X, right);
  curved_connect.llft("y", U, Y, dir -80);

  drawoptions(dashed withdots scaled 1/2);
  connect("(x,y)", U, XY);
  drawoptions();
endfig;
\end{mplibcode}
\end{document}