Let $Ring$ be the category of commuative unitary rings, let $k$ be a commutative unitary ring, $(Mod_{k},\otimes_{k},k)$ be the monoidal category of $k$-modules, $(Ab,\otimes_{\mathbb{Z}},\mathbb{Z})$ be the monoidal category of commutative groups, $Cat^{Mod_{k}}$ be the category of enriched category in $Mod_{k}$ and $Cat^{Ab}$ be the category of enriched category in $Ab$ respectively.
Let us begin recalling some fact: There exists a functor $\underline{(-)}:Ring\to Cat^{Ab}:R\to \underline{R} $, where $\underline{R}$ is the $Ab$-enriched category with only one object $*$ and whose $Ab$-hom-object is $\underline{R}(*,*)=R$.
It is well known that the triple $(Cat^{Ab},\otimes^{Ab},\underline{\mathbb{Z}})$ forms a monoidal category (see Riehl Categorical Homotopy Theory Section 7.3), whose tensor is defined as follow:
$-\otimes^{Ab} -:Cat^{Ab}\times Cat^{Ab}\to Cat^{Ab}:(A,B)\mapsto A\otimes^{Ab}B $,
where $A\otimes^{Ab}B$ is the enriched category whose objects are pair of objects of the starting enriched categories $(a,b)\in ob(A)\times ob(B)$ and whose Ab-hom-objects are, let $(a,b),(a_1,b_1)$ be a pair of objects of $A\otimes^{Ab}B$, $A\otimes^{Ab}B((a,b),(a_1,b_1))=A(a,a_1)\otimes_{\mathbb{Z}}B(b,b_1)$.
We need another category. Let $LMod_{\underline{k}}(Cat^{Ab})$ be the category of left $\underline{k}$-module objcets of $Cat^{Ab}$, whose objects are pair $(A,m_{A}:\underline{k}\otimes^{Ab} A\to A)$ composed by an object of $Cat^{Ab}$ as first component and as second component an action of the internal monoid $\underline{k}$ (with the obviously multiplication) on $A$ ( the action is a $Ab$-enriched functor $m_{A}:\underline{k}\otimes^{Ab} A\to A$ with the usual axioms) and whose arrows are arrows of $Cat^{Ab}$ which respect the action (i.e. $\underline{k}$-equivariant arrows).
It is well known that the extention-restriction of scalar adjunction
$L:Cat^{Ab}\rightleftarrows LMod_{\underline{k}}(Cat^{Ab}):R$.
is monadic.
Finally I can ask my question: is there an equivalence of categories $Cat^{Mod_{k}}\cong LMod_{\underline{k}}(Cat^{Ab})$?
My guess is yes and, I suspect, that this can be proved as follows. The extention-restrinction of scalar adjunction $\otimes_{\mathbb{Z}}k: Ab\rightleftarrows Mod_{k}:U$ is strong monoidal and then it induces an adjunction $-\otimes_{\mathbb{Z}}k^{*}: Cat^{Ab}\rightleftarrows Cat^{Mod_{k}}:U^{*}$.
Claim: Using monadicity theorem (https://ncatlab.org/nlab/show/monadicity+theorem), it is possible proves that the adjunction $-\otimes_{\mathbb{Z}}k^{*}\dashv U^{*}$ is monadic.
Assumimg the claim is true, we have finished. Indeed, the adjuctions $-\otimes_{\mathbb{Z}}k^{*}\dashv U^{*}$ and $L\dashv R$ are both monadic and both induce the same monad $T= \otimes_{\mathbb{Z}}\underline{k}:Cat^{Ab}\to Cat^{Ab}$, so their Eilenberg-Moore Algebras categories are equivalent.
I have never been able to prove the Claim. I have proved that $U^{*}$ is conservative, but I'm struggling to prove that $U^{*}$ preserves $U$-split coequalized.
I suspect that this result is in the literature, and if so, I would appreciate a refernece