Pushouts and Coequalizer

Question

Is it always the case that a pushout is a coequalizer of two arrows with the same domain and codomain? Some define coequalizers as a pushout and a coproduct with operations involving the universal arrows. What is the difference?

Answer 1

Ad definitions, pushout is just colimit of a diagram of shape “$• \leftarrow • \rightarrow •$” whereas coequalizer is colimit of a diagram of shape “$• \rightrightarrows •$”. So formally pushout cannot be equal to coequalizer (since colimit is an object plus collection of morphisms indexed by diagram). However there are ways of constructing colimits of diagrams of certain shape using colimits of diagrams of other shapes, for example to construct a pushout, first add coproduct of the two target objects to the diagram and then add coequalizer of the two oriented paths. The two compositions of added morphisms form the pushout.