ct.category theory – Is there a monoidal analogue of equalizers?

There are three different kinds of finite limits in categories: terminal objects, binary products, and equalizers. In a category $C$, these define functors $1_{C},times,mathrm{Eq}colonmathrm{Fun}(I,C)to C$ where $I=emptyset,{bullet bullet}$, and ${bulletrightrightarrowsbullet}$ respectively.

Monoidal categories generalise the first two in that we now have a functor $1_{C}$ from $mathcal{C}^{emptyset}=*$ to $mathcal{C}$ and a functor $otimes_C$ from $C^{{bullet bullet}}=C$ to $C$, i.e. functors
begin{align*}
1_C &colon * to mathcal{C}\
otimes_C &colon mathcal{C}timesmathcal{C} to mathcal{C}
end{align*}

together with associativity and unitality natural isomorphisms satisfying compatibility conditions.

What about equalizers? Has the notion of a category $C$ equipped with a functor $rm{Eq}colonrm{Fun}({bulletrightrightarrowsbullet},C)to C$, a unit functor, and unitality/associativity natural isomorphisms satisfying coherence conditions been studied before? Moreover, are there any examples of such structures “found in nature”?