Explicit unfurling for covariantly left adjointable functors

Develop an explicit, intrinsic description of the universal 2-functor U: Span(C, C_L, C)^{co} -> Cat_{∞} that extends a covariantly left L-adjointable functor F: C -> Cat_{∞} along the inclusion C -> Span(C, C_L, C)^{co}, analogous to the unfurling constructions given for the covariantly right L-adjointable and contravariantly adjointable cases, but without assuming that the cocartesian unstraightening Un^{cc}(F) has any cartesian edges or that the left adjoints f_! of F admit right adjoints.

Background

The paper proves that Barwick’s unfurling construction, when enhanced $(\infty,2)$-categorically, implements the universal property of span 2-categories for right adjointable and contravariantly adjointable functors. In these cases, the universal extensions are described concretely via straightening/unstraightening of 1-(co)cartesian fibrations of span 2-categories built from the cartesian or cocartesian unstraightening of the original functor.

However, for covariantly left L-adjointable functors F: C -> Cat_{∞}, the authors do not have a similarly explicit unfurling description. The naive pattern-matching expression Str^{ct}(Span(Un^{cc}F, cart, all)^{co}) fails because Un^{cc}(F) need not provide cartesian lifts, and even embedding F into a functor \hat F whose structure maps have right adjoints does not yield an intrinsic description depending only on Un^{cc}F. This leaves open a precise construction of the universal extension in this case.