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This is just an extension of Todd's answer to summarize Adamek's example, which is a bit convoluted. By a graph we mean a set equipped with a binary relation, call it ∼. If A is a graph, let $A^{(3)}$ be the set of triples (x,y,z) such that x∼y∼z. And let F(A) be the power set $P(A^{(3)})$, equipped with the relation defined by ∅∼X for all nonempty X. Adamek's category is the category of algebras for the endofunctor F of the category of graphs, i.e. of graphs A equipped with a graph-morphism F(A)→A. He proves that the forgetful functor from this category to Graphs has a left adjoint, namely $A\mapsto A \sqcup F(A)$, and is monadic. But he gives the following example of a pair of parallel morphisms of F-algebras that have no coequalizer.