2 Added info about effective equivalence relations.

As Finn says, lfp categories have all coequalizers. However, there is a fly in the ointment. In set-models of algebraic theories, the underlying functor preserves the congruences and the quotients of the congruences. This means that a congruence is an equivalence relation on the underlying set and the quotient alegbraic structure has underlying set the set-quotient of the congruence. This is true even in many sorted algebraic theories.

But in models of finite limit sketches the quotient can blow up. The quotient need not be a structure whose underlying set is the quotient of the equivalence relation on the underlying set. An example of this happens in the category of small categories when you merge nonisomorphic objects. All of a sudden arrows may compose that didn't meet each other before, creating new arrows. (So the underlying set of arrows in the quotient is not the quotient of the congruence on the underlying set.)

This is spelled out and proved in Toposes, Triples and Theories, by Michael Barr and Charles Wells, in Theorem 4.1 of Chapter 8. In that theorem, "LE" means "Finite Limits" and "EE" means "effective equivalence relations whose quotients are preserved by the underlying set functor". This book is available for free on the internet -- just google it. Exercises EEPO and ORTHODOX give specific examples of that behavior. I think there is a specific example for small categories somewhere but at the moment I can't find it.

ADDED: The specific example is in section 1.8, exercise CBB. Section 1.8 talks about effective equivalence relations in general and calls them congruences.

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As Finn says, lfp categories have all coequalizers. However, there is a fly in the ointment. In set-models of algebraic theories, the underlying functor preserves the congruences and the quotients of the congruences. This means that a congruence is an equivalence relation on the underlying set and the quotient alegbraic structure has underlying set the set-quotient of the congruence. This is true even in many sorted algebraic theories.

But in models of finite limit sketches the quotient can blow up. The quotient need not be a structure whose underlying set is the quotient of the equivalence relation on the underlying set. An example of this happens in the category of small categories when you merge nonisomorphic objects. All of a sudden arrows may compose that didn't meet each other before, creating new arrows. (So the underlying set of arrows in the quotient is not the quotient of the congruence on the underlying set.)

This is spelled out and proved in Toposes, Triples and Theories, by Michael Barr and Charles Wells, in Theorem 4.1 of Chapter 8. In that theorem, "LE" means "Finite Limits" and "EE" means "effective equivalence relations whose quotients are preserved by the underlying set functor". This book is available for free on the internet -- just google it. Exercises EEPO and ORTHODOX give specific examples of that behavior. I think there is a specific example for small categories somewhere but at the moment I can't find it.