Equate the enriched-kernel 2-category with the span 2-category

Establish that for any category C with finite limits, the canonical 2-functor f:(C)→^enr(C) obtained by changing the enrichment of the span 1-category Span(C) along the lax monoidal functor x↦Span(C_{/x}) (so that the underlying 1-category of ^enr(C) is Span(C) and C→^enr(C) is biadjointable) is an equivalence of 2-categories. Concretely, prove that the natural equivalence of hom-categories _^{enr(C)}(x,y)≃Span(C_{/x×y})≃Span(C_{/x}×_C C_{/y}) is functorial in both variables x and y, thereby implying that f is fully faithful and essentially surjective and giving a rigorous identification (C)≃^enr(C) with the same universal biadjointable property.

Background

The paper constructs the span 2-category #4,#3{C}{E}{I}{P} and proves its universal property as the initial (I,P)-biadjointable 2-functor extension of C. It also sketches an alternative approach to constructing the universal biadjointable functor using categories of kernels and enrichment transfer, yielding a Cat-enriched category enr(C). This construction provides a 2-functor f:(C)→enr(C) that is the identity on objects and whose hom-categories are abstractly equivalent to Span(C_{/x×y}).

While this strongly suggests that f should be an equivalence of 2-categories, the authors note that a rigorous proof would require demonstrating that these hom-equivalences are functorial in x and y, which in turn needs results about the interaction between transfer of enrichments and the enriched Yoneda embedding—tools that the authors say do not appear to be available in the literature yet.

References

There is also another natural approach one could try to use to construct the universal biadjointable functor; we briefly sketch this construction, and then explain why it is not clear—albeit quite likely—that this has the correct universal property. This makes it seem very likely that this 2-functor is in fact an equivalence, but is not yet a rigorous proof. In order to show this, note that it suffices by our recognition principle to make the equivalence eq:hom_in_enr functorial in $x \in C$. While the functoriality in $(#1 C)$ is automatic from the construction of $\text{enr}(C)$, we expect that the full functoriality in $x$ would require a discussion of the interaction between transfer of enrichments and the enriched Yoneda embedding, which does not yet seem to have appeared in the literature.

eq:hom_in_enr:

$_{^{\text{enr}}}(x,y)\simeq\Span(C_{/x\times y})\simeq\Span(C_{/x}\times_CC_{/y}) $

— Universality of span 2-categories and the construction of 6-functor formalisms (2505.19192 - Cnossen et al., 25 May 2025) in Remark, Section 4.2 (Universality of span 2-categories; after Theorem 4.1)