The sliding rule seems to be: you can compose tensors with different dimensions by sticking together terms with same components which is essentially following morphisms down the line. The cups, caps and morphisms can be composed on either one of its ends. Identity morphisms can always disappear and reappear around same subscripts.

Using your equations for \$$g^\ast f^\ast\$$, if we collect terms we get:

$\cap_x\ \cdot 1_x \cdot 1_x \otimes g \cdot (\cap_y \cdot 1_y \cdot \cup_y) \cdot f \otimes 1_z \cdot 1_z \cdot \cup_z$
$\cap_x\ \cdot 1_x \cdot 1_x \otimes g \cdot (1_y \cdot \cap_y) \cdot (\cup_y \cdot 1_y) \cdot f \otimes 1_z \cdot 1_z \cdot \cup_z$
$\cap_x\ \cdot 1_x \cdot 1_x \otimes g \cdot f \otimes 1_z \cdot 1_z \cdot \cup_z$