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Let [math]\mathbf{C}[/math] be a category and [math]F : \mathbf{C} \to \mathbf{C}[/math] an endofunctor on [math]\mathbf{C}[/math]. An [math]F[/math]-coalgebra is a morphism [math]c : X \to F X[/math] in [math]\mathbf{C}[/math]. A homomorphism from a coalgebra [math]c : X \to F X[/math] to [math]d : Y \to F Y[/math] is a morphism [math]f : X \to Y[/math] such that [math]F f \circ c = d \circ f[/math].

<theorem id="coalg-cat">%c  

 [math]F[/math]-Coalgebras and their homomorphisms form a category [math]\mathrm{Coalg}(F)[/math].