  We show that the three different looking BPS partition functions, namely the
elliptic genus of the 6d $\mathcal{N}=(1,0)$ $Sp(1)$ gauge theory with $10$
flavors and a tensor multiplet, the Nekrasov partition function of the 5d
$\mathcal{N}=1$ $Sp(2)$ gauge theory with $10$ flavors, and the Nekrasov
partition function of the 5d $\mathcal{N}=1$ $SU(3)$ gauge theory with $10$
flavors, are all equal to each other under specific maps among gauge theory
parameters. This result strongly suggests that the three gauge theories have an
identical UV fixed point. Type IIB 5-brane web diagrams play an essential role
to compute the $SU(3)$ Nekrasov partition function as well as establishing the
maps.
