The equalizer of a collection of morphisms is the limit of the diagram containing these morphisms. cat
Thus for any ,
and given any other morphism with this property there exists a unique such that the following diagram, except for , commutes
Note that in case we take the diagram consisting of only .
Thus the equalizer is the “most general” subobject for which the morphisms concur.
The coëqualizer of a collection of morphisms is the colimit of the diagram containing these morphisms. cat
Thus ,
and given any other morphism there exists a unique such that the following diagram commutes, except for :
Note that in case we take the diagram consisting of only .
Thus the coëqualizer is the “most general” quotient object onto which the morphisms concur.