# are comma-objects semi-coflexible

I need someone to read through my proof, because I feel very uncertain about 2-categorical limits. A strict indexed category $$C:\mathscr S^{op}\to Cat$$ is semi coflexible when every pseudo-transformation with codomain $$C$$ is isomorphic to a strict one. This is equivalent to saying that the action of the forgetful 2-functor $$U:2Cat_s(A,C) \to 2Cat(UA,UC)$$ is an equivalence for each strict indexed category $$A$$.

Definition. (of comma objects)

Let $$K$$ be any 2-category. The comma-object $$f/g$$ of two 1-cells $$f:D\to C$$ and $$g:B\to C$$ is a diagram as the one shown below

such that for any 0-cell $$A$$ the functor which sends $$u:A\to f/g$$ to the triple $$(p_1u,p_2u, \theta u)$$ and a 2-cell $$\sigma: u\Rightarrow v$$ to the pair $$(p_1 \sigma, p_2 \sigma)$$ is an equivalence of $$K(A,f/g)$$ with the category whose objects are pairs $$(a,b,\gamma)$$ consisting of two cells $$a:A\to D$$ and $$b:A\to B$$ and a transformation $$\gamma :fa\Rightarrow gb$$ and whose morphisms $$(\alpha,\beta):(a,b,\gamma)\to (a',b',\gamma')$$ consits of pairs of transformations $$\alpha:a\Rightarrow a'$$ and $$\beta:b\Rightarrow b'$$ such that $$\gamma' \circ (f\alpha) =(g\beta)\circ \gamma$$. I call that the category of lax cones above the span (even though it isn't exactly that category, because the arrows into $$C$$ are missing).

Claim. (that I want to prove) When $$f$$ and $$g$$ as above are 1-cells in the 2-category $$2Cat_s(\mathscr S^{op},Cat)$$ of strict indexed categories, strict transformations and modifications, and when $$D$$, $$B$$ and $$C$$ are all semi coflexible, then so is the comma object $$f/g$$.

Proof. I need to check that for each $$A$$ the functor $$2Cat_s(A,f/g)\to 2Cat(UA,U(f/g))$$ is an equivalence of categories. First note that applying $$U$$ to everything in the diagram above gives me something which again satisfies the universal property of a comma square in the 2-category $$2Cat(\mathscr S^{op},Cat)$$. This is because $$U$$ has a left 2-adjoint and should hence preserves lax limits. Thus I see that $$2Cat(UA,U(f/g))\simeq 2Cat(UA,Uf/Ug)$$ is equivalent to the category of lax cones above the span $$Uf:UD\to UC\leftarrow UB:Ug$$. The category $$2Cat_s(A,f/g)$$ is equivalent to the category of lax cones above $$f:D\to C\leftarrow B:g$$. The 2-functor $$U$$ induces a functor from the latter to the former 2-category of lax cones, and now using that every 1-cell $$UA\to UD$$ and $$UA\to UB$$ can be replaced by an isomorphic strict one I see that the functor induced by $$U$$ between the categories of lax cones is essentially surjective on objects. To see that is is fully faithful I use that $$U$$ itself is locally fully faithful. It follows that $$2Cat_s(A,f/g) \to 2Cat(UA,U(f/g))$$ is an equivalence for each strict $$A$$, and $$f/g$$ is semi-coflexible.

Are there any obvious red flags?