  An important part of a CFT four-point function, the stress tensor sector,
comprises the exchanges of the stress tensor and its composites. The OPE
coefficients of these multi-stress tensor operators and consequently, the
complete stress tensor sector of four-point functions in CFTs with a large
central charge, can be determined by computing a heavy-heavy-light-light
correlator. We show how one can make substantial progress in this direction by
bootstrapping a certain ansatz for the stress tensor sector of the correlator,
iteratively computing the OPE coefficients of multi-stress tensor operators
with increasing twist. Some parameters are not fixed by the bootstrap - they
correspond to the OPE coefficients of multi-stress tensors with spin zero and
two. We further show that in holographic CFTs one can use the phase shift
computed in the dual gravitational theory to reduce the set of undetermined
parameters to the OPE coefficients of multi-stress tensors with spin zero.
Finally, we verify some of these results using the Lorentzian OPE inversion
formula and comment on its regime of applicability.
