Example [efr-0039]

Let C: \Delta ^\mathrm {op} \to \mathsf {Cat} be a weak Segal category. Given two compatible horizontal morphisms---that is, two objects f,g \in C[1], so that d_0(f) = d_1(g), this (coupled with the identity morphism on this object of C[0]) describes an object in the weak pullback C[1] \times _{C[0]}^\sim C[1]. This object is in the essential image of C[2] \to C[1] \times _{C[0]}^\sim C[1], (since it is an equivalence, hence essentially surjective,) but not necessarily the image. This means we can't necessarily find a composite f;g with the same domain as f, for example---the best we can do is ask for the domain to be vertically isomorphic to d(f).