Cotensor products (in monoidal categories) without regularity 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 $M\boxtimes_{D}N$ defined by

To define the left $C$-and right $E$-coactions of $M\boxtimes_{D}N$, however, Aguiar assumes that $\mathcal{V}$ is regular, i.e. that it has all equalisers and, for each parallel pair of morphisms $f,g\colon X\rightrightarrows Y$ of $\mathcal{V}$ and each $A,B\in\mathrm{Obj}(\mathcal{V})$, we have
$$
A\otimes_{\mathcal{V}}\mathrm{Eq}(f,g)\otimes_{\mathcal{V}}B
\cong
\mathrm{Eq}(\mathrm{id}_{A}\otimes_{\mathcal{V}}f\otimes_{\mathcal{V}}\mathrm{id}_{B},\mathrm{id}_{A}\otimes_{\mathcal{V}}g\otimes_{\mathcal{V}}\mathrm{id}_{B}).
$$
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 $M\boxtimes_{D}N$ 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 $M\boxtimes_{D}N$ has the structure of a $(C,E)$-bicomodule in that we have coactions of the form
$$
\begin{align*}
    \alpha^{\mathrm{L}}_{M\boxtimes_{D}N} &\colon M\boxtimes_{D}N \longrightarrow C\otimes_{\mathcal{V}}(M\boxtimes_{D}N),\\
    \alpha^{\mathrm{R}}_{M\boxtimes_{D}N} &\colon M\boxtimes_{D}N \longrightarrow (M\boxtimes_{D}N)\otimes_{\mathcal{V}}E
\end{align*}
$$
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
$$
\alpha
\colon
(M\boxtimes_{D}N)\boxtimes_{E}P
\overset{{\!\sim}}{\dashrightarrow}
M\boxtimes_{D}(N\boxtimes_{E}P).
$$

*(Unitors (Proposition 2.2.3 of Aguiar's Thesis): We have coherent isomorphisms $C\boxtimes_{C}M\cong M$ and $M\boxtimes_{D}D\cong M$. This holds without regularity already.)

 A: First, notice that the whole situation can be dualized by passing to the dual monoidal category (which is perhaps more intuitive to understand). So we are looking at bimodules over monoid objects in a monoidal category, and want to construct their tensor product. That tensor product should be defined by a universal property (very similar to the definition of a coend). The coequalizer then provides a construction of that universal object (but it is not the definition). The monoidal category should have reflexive coequalizers, and of course the tensor product should be compatible with these in both variables (for comonoids and cobimodules: coreflexive equalizers). Without such an assumption it is very unlikely to get a satisfactory construction of tensor products of bimodules, even of modules over commutative monoid objects in the symmetric monoidal case, which then can be seen as algebras for a monoidal monad. The paper Tensors, monads and actions by Gavin Seal is a good read on that. Chapter 6 in my thesis also deals with this kind of situation. Generally speaking, monads preserving reflexive coequalizers are much better behaved than general monads (thus, comonads are better when they preserve coreflexive equalizers), this has already been shown by Linton in Coequalizers in categories of algebras. Coming back to bimodules, they can be seen as algebras for the composition of two monoidal monads equipped with a distributive law, so the monad case is more or less the general one.
