Equivalence of several descriptions for 6d SCFT

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.