  We introduce the triangular prism equations for fusion categories, which turn
out to be equivalent to the pentagon equations in the spherical case (up to a
change of basis), but provide insight to manage the complexity by localization.
We also prove a conjecture of Z. Wang about the second Frobenius-Schur
indicator on pivotal fusion categories. As applications, we provide new
categorification criteria and complete the classification of the unitary simple
integral fusion categories of Frobenius type up to rank 8 and FPdim 4080.
