  We consider a natural generalization of the Carlsson-Okounkov Ext operator on
the K-theory groups of the moduli spaces of stable sheaves on a smooth
projective surface. We compute the commutation relations between the Ext
operator and the action of the deformed W-algebra on K-theory, which was
developed in [18]. The conclusion is that the Ext operator is closely related
with a vertex operator, thus giving a mathematical incarnation of the
Alday-Gaiotto-Tachikawa correspondence for a general algebraic surface.
