  Models of compositeness can successfully address the origin of the Higgs
boson, as a pseudo Nambu Goldstone boson (pNGB) of a spontaneously broken
global symmetry, and flavour physics via the partial compositeness mechanism.
If the dynamics is generated by a confining gauge group with fermionic matter
content, there exists only a finite set of models that have the correct
properties to account for the Higgs and top partners at the same time. In this
letter we explore the theory space of this class of models: remarkably, all of
them contain - beyond the pNGB Higgs - a pNGB singlet, $a$, which couples to
Standard Model gauge bosons via Wess-Zumino-Witten interactions, thus providing
naturally a resonance in di-boson at the LHC. With the assumption that the
recently reported di-photon excess at 750 GeV at the LHC arises from the
a-resonance, we propose a generic approach on how to delineate the best
candidate for composite Higgs models with top-partners. We find that
constraints from other di-boson searches severely reduce the theory space of
the models under consideration. For the models which can explain the di-photon
excess, we make precise and testable predictions for the width and other
di-boson resonance searches.
