According to the wikipedia entry on Categorical Logic:
Lawvere's writings, sometimes couched in a philosophical jargon, isolated some of the basic concepts as adjoint functors (which he explained as 'objective' in a Hegelian sense, not without some justification).
Hegel had continued with Kants bridging of the world of the subject & object, but whereas Kant mentions antinomies only in passing to point out the limits of experience, Hegel makes this the engine that drives his philosophy through his triadic dialectic.
Lawvere is known for advocating the adjunction between the space of logics and the space of its concrete models. Possibly he considered this a motion between the world of thought and that of concrete reality. But there is no sense of a dialectic here in the Hegelian sense.
Is there a well-known way that Hegel used the term 'objective'? For example, I've come across 'Objective Spirit'.