Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?

Question

I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we additionally require that the fixed-point spectra have actions, we get a sheaf with transfer. This formalism can be applied to describe equivariant (spectral) stacks based on arbitrary topoi. I'm wondering: is this the strongest reasonable notion of gluing we can expect here? I would think the transfers ought to satisfy their own gluing formula, so that we retain the duality inherent to Mackey functors. Do transfers automatically have a (co)gluing property? Or is there a reasonable one to impose?