In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations.

A congruence relation is an equivalence relation $\sim$ on $A$ such that the operations on $A$ produce well-defined operations on the set $A/\sim$ of equivalence classes by applying them to representatives. I.e. for any operation $f$ on $A$ the operation $\bar{f}$ on $A/\sim$ given by $\bar{f}([x_1],...,[x_n]):=[f(x_1,...x_n)]$ is well-defined, i.e. if $x_1 \sim y_1, ... , x_n \sim y_n$ then $f(x_1,...,x_n) \sim f(y_1,...,y_n)$ for all operations $f$ of the given structure. Thus these relations are the right ones to form quotients inside the given category of algebraic structures.

An essentially algebraic structure is a (if 1-sorted) or several (if many-sorted) sets with partially defined operations satisfying equational laws, where the domain of any given operation is a subset defined by equations between previously defined operations (equivalenty: it is a $Set$-model of a finite limit sketch). The standard example are categories, where one has three global operations, identity, source and target, and a partial operation, composition, defined only for certain pairs of morphisms.

My question is: Is there a notion of congruence relation for these more general algebraic structures? E.g. one equivalence relation on each set satisfying the analogous properties to the above? If so have these "congruence relations" been studied, do they e.g. form lattices?

Motivation: Just curiosity really. I asked myself this question, after reading this MO-question of Colin Tan, which might be a special case. He asks whether there is a way to collapse two objects in a category. If there was a lattice of congruence relations on a category, there might be the congruence relation generated by the relation which identifies just the two objects (this would of course mean to treat categories in an "evil", non-two-categorical way, but that was what the question sounded like to me). Googling did reveal nothing, so I ask you people...

In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations.

A congruence relation is an equivalence relation $\sim$ on $A$ such that the operations on $A$ produce well-defined operations on the set $A/\sim$ of equivalence classes by applying them to representatives. I.e. for any operation $f$ on $A$ the operation $\bar{f}$ on $A/\sim$ given by $\bar{f}([x_1],...,[x_n]):=[f(x_1,...x_n)]$ is well-defined, i.e. if $x_1 \sim y_1, ... , x_n \sim y_n$ then $f(x_1,...,x_n) \sim f(y_1,...,y_n)$ for all operations $f$ of the given structure. Thus these relations are the right ones to form quotients inside the given category of algebraic structures.

An essentially algebraic structure is a (if 1-sorted) or several (if many-sorted) sets with partially defined operations satisfying equational laws, where the domain of any given operation is a subset defined by equations between previously defined operations (equivalenty: it is a $Set$-model of a finite limit sketch). The standard example are categories, where one has three global operations, identity, source and target, and a partial operation, composition, defined only for certain pairs of morphisms.

My question is: Is there a notion of congruence relation for these more general algebraic structures? E.g. one equivalence relation on each set satisfying the analogous properties to the above? If so have these "congruence relations" been studied, do they e.g. form lattices?

Motivation: Just curiosity really. I asked myself this question, after reading this MO-question of Colin Tan, which might be a special case. He asks whether there is a way to collapse two objects in a category. If there was a lattice of congruence relations on a category, there might be the congruence relation generated by the relation which identifies just the two objects (this would of course mean to treat categories in an "evil", non-two-categorical way, but that was what the question sounded like to me). Googling did reveal nothing, so I ask you people...