Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have isomorphisms $\varphi_0:\Bbbk\to\omega(\mathbb{I})$ and $\varphi=\left(\varphi_{X,X}:\omega(X)\otimes \omega(Y)\to\omega(X\otimes Y)\right)_{X,Y\in\mathcal{C}}$ but these are not necessarily compatible with the constraints.
In light of Majid's Tannaka-Krein theorem for quasi-Hopf algebras and other results we know that there exists a coquasi-bialgebra $H$ (namely, $H=\mathsf{coend}(\omega)$) such that $\omega$ factors through the forgetful functor $\mathcal{U}:\mathsf{Com}^{H}_{fd}\to\mathsf{vec}_{\Bbbk}$ from the category of finite-dimensional $H$-comodules to $\mathsf{vec}_{\Bbbk}$, that is to say, there exists a functor $\mathcal{I}_{\mathcal{C}}:\mathcal{C}\to \mathsf{Com}^{H}_{fd}$ such that $\mathcal{U}\mathcal{I}_{\mathcal{C}} = \omega$.
I have been told that:
Claim: Since any object in $\mathsf{Com}^{H}_{fd}$ can be obtained from objects in the image of $\mathcal{I}_{\mathcal{C}}$ by taking direct sums, kernel and cokernels and since in an abelian monoidal category with exact tensor product (as $\mathsf{Com}^{H}_{fd}$) the collection of rigid objects is closed under taking sums, kernels and cokernels, the category $\mathsf{Com}^{H}_{fd}$ has to be rigid as well.
My main question is:
Main Question: Is this true?
It is since a long time that I am trying to prove this claim, but I couldn't manage to.
In light of Schauenburg's Tannaka duality for arbitrary Hopf algebras, Corollary 2.2.9, every object in $\mathsf{Com}^{H}_{fd}$ is the quotient of a subobject of a finite biproduct of objects from the image of the functor $\mathcal{I}_{\mathcal{C}}$, but this is not the same as Claim, whence I provided my own proof of the fact that any object in $\mathsf{Com}^{H}_{fd}$ can be obtained from objects in the image of $\mathcal{I}_{\mathcal{C}}$ by taking direct sums, kernel and cokernels. It is quite long and technical, but I think it should work, whence I'll omit it.
My problem now is with the second part of the Claim, about abelian monoidal categories. Since I didn't manage to deal with it directly, I am trying to approach the problem step by step.
Let $\Bbbk$ be a von Neumann regular commutative algebra (i.e. every $\Bbbk$-module is flat) and let $\mathsf{Mod}_{\Bbbk}$ be the category of $\Bbbk$-modules. The collection of rigid objects here should coincide with the finitely-generated and projective modules. Thus, if $f:M\to N$ is a morphism of fgp $\Bbbk$-modules:
Sub-question 1: is it true that $\ker(f)$ is fgp as well?
Sub-question 2: what about $\operatorname{coker}(f)$?
Any help or comment (even rude ones) will be very welcome.
NB: this question is strictly related with this other question on MSE, which however didn't receive an answer up to know.