Does the bicategory $\bf Prof$ of categories, profunctors and natural transformations admit all pseudolimits?


By [Kel89, Prop. 5.1] it is enough to show that $\bf Prof$ admits products, cotensors, iso-inserters and equifiers. It's easy to show that products and cotensors exist, but what about inserters and equifiers?


Is there a source I can cite or a more general theorem ensuring this is true (or maybe a counterexample?)?


[Kel89] G.M. Kelly, *Elementary observations on 2-categorical limits*



Does $\bf Prof$ admit all pseudolimits?
There is another simple kind of peusdo-limits called inverters, in which one universally inverts a 2-morphism between two parallel 1-morphisms (such pseudo-limits can also be constructed in a simple way from iso-inserters and equifiers). We now claim that ${\bf Prof}$ does **not** have all inverters. 


To construct a counterexample, let ${\bf C}$ be a small, non-empty and idempotent complete category, and let $K: {\bf C} \looparrowright {\bf C}$ be a pro-functor which is given by a subfunctor $K(-,-) \subseteq {\rm Hom}(-,-): {\bf C^{\rm op}} \times {\bf C} \to {\bf Set}$ of the Hom set functor. We may consider $K$ as a collection of morphisms in ${\bf C}$ which is closed under pre-composition and post-composition by arbitrary morphisms in ${\bf C}$. Given an object $X \in {\bf C}$ let us denote by $I\_X \subseteq {\bf C}\_{/X}$ the full subcategory spanned by those arrows $f: Z \to X$ which are in the image of the inclusion $K(Z,X) \subseteq {\rm Hom}(Z,X)$. Similarly, if $f: X \to Y$ is in the image of $K(X,Y)$ then we will denote by $I\_f \subseteq ({\bf C}\_{X/})\_{/f}$ the full subcategory spanned by those factorizations $X \stackrel{f'}{\to} Z \stackrel{f''}{\to} Y$ of $f$ such that $f'$ and $f''$ belong to the image of $K$. Let us now assume that $K$ satisfies the following properties:


(1) For every morphism $f: X \to Y$ in $K$, the category of factorizations $I\_f$ is weakly contractible.


(2) For every $X \in {\bf C}$ the natural diagram $F\_X:I\_X^{\triangleright} \to {\bf C}$ with cone $X$ is a colimit diagram.


(3) The identity ${\rm Id}: X \to X$ does not belong to $K(X,X)$ for any $X \in {\bf C}$.


Let us now assume we found such a ${\bf C}$ and such a $K$. Consider the diagram in ${\bf Prof}$ consisting of the two profunctors $K,{\rm Id}: {\bf C} \looparrowright {\bf C}$ and the 2-morphism $\epsilon: K \Rightarrow {\rm Id}$ determined by the inclusion $K(-,-) \subseteq {\rm Hom}(-,-)$. We claim that this diagram does not admit an inverter in ${\bf Prof}$. Suppose otherwise and let $\iota:{\bf D} \looparrowright {\bf C}$ be the inverter. Applying the functor ${\bf Prof}(\ast,-)$ to our inverter diagram we obtain an diagram of categories identifying the functor category ${\bf Set}^{\bf D}$ as the full subcategory of ${\bf Set}^{\bf C}$ spanned by those functors $f: {\bf C} \to {\bf Set}$ such that $\epsilon\_\*(f):K\_\*(f) \to f$ is an isomorphism, where
$K\_\*: {\bf Set}^{\bf C} \to {\bf Set}^{\bf C}$ is the functor induces by post-composition with $K$ and $\epsilon\_\*: K\_\* \Rightarrow {\rm Id}$ is the induced natural transformation. Unwinding the definitions we see that $K\_\*(f): {\bf C} \to {\bf C}$ is given by the formula $K\_\*(f)(X) = {\rm colim}\_{Y \to X \in I\_X}f(Y)$ and $\epsilon\_\*(f)$ is induced by the diagram $F\_X:I\_X^{\triangleright} \to {\bf C}$ with cone $X$. In other words, the inverter diagram identifies ${\bf Set}^{\bf D}$ with the full subcategory ${\rm Fun}\_{\{F\_X\}}({\bf C},{\bf Set}) \subseteq {\bf Set}^{\bf C}$ spanned by those of functors ${\bf C} \to {\bf Set}$ which preserve all the colimit diagrams $F\_X$. We will now reach a contradiction by showing that ${\rm Fun}\_{\{F\_X\}}({\bf C},{\bf Set})$ does not admit any continuous cocontinuous functors to ${\bf Set}$, and in particular cannot be of the form ${\bf Set}^{\bf D}$ for any small category ${\bf D}$ (unless ${\bf D}$ is empty, but this case can be ruled out easily in any given example).


Condition (1) above implies that the induced natural transformations $\epsilon\_\*\circ K\_\*,K\_\*\circ \epsilon\_\*: K\_\* \circ K\_\* \to K\_\*$ are natural isomorphisms. In particular, $K\_\*$ is a **colocalization functor** and hence factors as $${\bf Set}^{\bf C} \stackrel{K\_\*'}{\to} {\rm Fun}\_{\{F\_X\}}({\bf C},{\bf Set}) \stackrel{\iota}{\to} {\bf Set}^{\bf C} $$
where $K\_\*'$ is right adjoint to $\iota$. In particular $K\_\*'$ preserves all limits. Since $K\_\*$ preserves all colimits and $\iota$ detects colimits it follows that $K\_\*'$ also preserves colimits. This means that if $F: {\rm Fun}\_{\{F\_X\}}({\bf C},{\bf Set}) \to {\bf Set}$ is a functor preserving both limits and colimits then $F \circ K\_\*': {\bf Set}^{\bf C} \to {\bf Set}$ preserves all limits and colimits and since we assumed ${\bf C}$ to be idempotent complete it follows that $F \circ K\_\*$ is given by an evaluation functor ${\rm ev}\_X: {\bf Set}^{\bf C} \to {\bf Set}$ for some $X \in {\bf C}$, and hence $F \cong F \circ K\_\* \circ \iota \cong {\rm ev}\_X \circ \iota$ is also given by evaluation at $X$. Now since the diagram $F\_X:I\_X^{\triangleright} \to {\bf C}$ is a colimit diagram it follows that $F^{\rm op}\_X:(I^{\rm op}\_X)^{\triangleleft} \to {\bf C^{op}}$ is a limit diagram and hence determines a limit diagram in ${\bf Set}^{\bf C}$ via the Yoneda embedding. Applying the functor $K\_\*$ we now get a limit diagram $G\_X:(I^{\rm op}\_X)^{\triangleleft} \to {\rm Fun}\_{\{F\_X\}}({\bf C},{\bf Set})$. Unwinding the definitions we see that $G\_X$ sends $Y \to X \in I^{\rm op}\_X$ to the functor $K(Y,-):{\bf C} \to {\bf Set}$ and sends the cone point to the functor $K(X,-)$. Since we assumed $F \cong {\rm ev}\_X \circ \iota$ preserves limits it follows that ${\rm ev}\_X\iota G\_X$ is a limit diagram in ${\bf Set}$, i.e., the natural map $K(X,X) \to {\rm lim}\_{Y \to X \in I^{op}\_X}K(Y,X)$ is an isomorphism of sets. On the other hand, the natural map ${\rm Hom(X,X)} \to {\rm lim}\_{Y \to X \in I^{op}\_X}{\rm Hom}(Y,X)$ is an isomorphism as well by assumption (1), which means that a map $g:X \to X$ belongs to $K(X,X)$ if and only if $g \circ f$ belongs to $K(Y,X)$ for every $f \in K(Y,X)$. But this is a contradiction to our assumption (3) which stated that ${\rm Id}: X \to X$ does not belong to $K(X,X)$.


Finally, let us indicate an example for a ${\bf C}$ and a $K$ which satisfy assumptions (1)-(3). Let $X$ be a non-empty connected compact Hausforff space with more then one point and let ${\bf C}$ be the poset of open subsets $U \subseteq X$ (where an arrow goes from the contained open set to the containing open set). Then ${\bf C}$ is a poset and hence idempotent complete. For open subsets $U,V \subseteq X$ let $K(U,V) \subseteq {\bf C}(U,V)$ be a singleton if the closure of $U$ is properly contained in $V$ (i.e. if $\overline{U} \subsetneq V$) and empty otherwise. It is then not hard to show that properties (1)-(3) are satisfied. 


