In *Internal Categories and Quantum Groups*, Aguiar defines the **cotensor product** of two bicomodules as follows. Let

- $(mathcal{V},otimes_{mathcal{V}},mathbf{1}_{mathcal{V}})$ be a monoidal category;
- $(C,Delta_{C},epsilon_{C})$, $(D,Delta_{D},epsilon_{D})$, and

$(E,Delta_{E},epsilon_{E})$ be comonoids in $mathcal{V}$;
- $(M,alpha^{mathrm{L}}_{M},alpha^{mathrm{R}}_{M})$ be a $(C,D)$-bicomodule in $mathcal{V}$;
- $(N,alpha^{mathrm{L}}_{N},alpha^{mathrm{R}}_{N})$ be a $(D,E)$-bicomodule in $mathcal{V}$.

Then the **cotensor product** of $M$ and $N$ is the $(C,E)$-bicomodule $Mboxtimes_{D}N$ defined by

To define the left $C$-and right $E$-coactions of $Mboxtimes_{D}N$, however, Aguiar assumes that $mathcal{V}$ is **regular**, i.e. that it has all equalisers and, for each parallel pair of morphisms $f,gcolon Xrightrightarrows Y$ of $mathcal{V}$ and each $A,Binmathrm{Obj}(mathcal{V})$, we have

$$

Aotimes_{mathcal{V}}mathrm{Eq}(f,g)otimes_{mathcal{V}}B

cong

mathrm{Eq}(mathrm{id}_{A}otimes_{mathcal{V}}fotimes_{mathcal{V}}mathrm{id}_{B},mathrm{id}_{A}otimes_{mathcal{V}}gotimes_{mathcal{V}}mathrm{id}_{B}).

$$

This in principle cuts out some desirable examples, as e.g. $mathsf{Mod}_{R}$ is regular iff all $R$-modules are flat (i.e. iff $R$ is von Neumann regular). On the other hand, I have seen it claimed elsewhere that, in the case of $mathsf{Mod}_{R}$, one *can* endow $Mboxtimes_{D}N$ with the structure of a $(C,E)$-bicomodule without any mention of $R$ being regular.

**Question.** Can one prove the following basic properties of bicomodules that Aguiar uses in his thesis without assuming $mathcal{V}$ is regular (or perhaps under weaker hypotheses)?

**Left and Right Coactions** (Proposition 2.2.1 of Aguiar’s Thesis)**:** The object $Mboxtimes_{D}N$ has the structure of a $(C,E)$-bicomodule in that we have coactions of the form

$$

begin{align*}

alpha^{mathrm{L}}_{Mboxtimes_{D}N} &colon Mboxtimes_{D}N longrightarrow Cotimes_{mathcal{V}}(Mboxtimes_{D}N),\

alpha^{mathrm{R}}_{Mboxtimes_{D}N} &colon Mboxtimes_{D}N longrightarrow (Mboxtimes_{D}N)otimes_{mathcal{V}}E

end{align*}

$$

satisfying the axioms of a $(C,E)$-bicomodule in $mathcal{V}$.
**Associators** (Proposition 2.2.2 of Aguiar’s Thesis)**:** For every triple $(M,N,P)$, we have a coherent isomorphism

$$

alpha

colon

(Mboxtimes_{D}N)boxtimes_{E}P

overset{{!sim}}{dashrightarrow}

Mboxtimes_{D}(Nboxtimes_{E}P).

$$
- (
**Unitors** (Proposition 2.2.3 of Aguiar’s Thesis)**:** We have coherent isomorphisms $Cboxtimes_{C}Mcong M$ ~~and~~ $Mboxtimes_{D}Dcong M$. *This holds without regularity already*.)