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?



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