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”?