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It will be seen that if $H$ is a weak Hopf algebra in the definition ofcoaction of weak bialgebras on coalgebras \cite{Wang}, then a definitionproperty is suppressed giving rise to the (global) coactions of weak Hopfalgebras on coalgebras. The next step will be introduce the more general notionof partial coactions of weak Hopf algebras on coalgebras as well as a family ofexamples via a fixed element on the weak Hopf algebra, illustrating bothdefinitions: global and partial. Moreover, it will also be presented how toobtain a partial comodule coalgebra from a global one via projections, givinganother way to find examples of partial coactions of weak Hopf algebras oncoalgebras. In addition, the weak smash coproduct \cite{Wang} will be studiedand it will be seen under what conditions it is possible to generate a weakHopf algebra structure from the coproduct and the counit defined on it.Finally, a dual relationship between the structures of partial action andpartial coaction of a weak Hopf algebra on a coalgebra will be established.

Partial Coactions of Weak Hopf Algebras on Coalgebras